Theorem oaun3 42814

Description: Ordinal addition as a union of classes. (Contributed by RP, 13-Feb-2025.)

Assertion

Ref	Expression
oaun3	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})

Distinct variable groups:   𝐴,𝑎,𝑏,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑥,𝑦,𝑧

Proof of Theorem oaun3

Step	Hyp	Ref	Expression
1		oacl 8560	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
2	1	difexd 5333	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) ∈ V)
3		uniprg 4926	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ((𝐴 +o 𝐵) ∖ 𝐴) ∈ V) → ∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} = (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)))
4	2, 3	syldan 589	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} = (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)))
5		undif2 4478	. . . . 5 ⊢ (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)) = (𝐴 ∪ (𝐴 +o 𝐵))
6		oaword1 8577	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝐵))
7		ssequn1 4180	. . . . . 6 ⊢ (𝐴 ⊆ (𝐴 +o 𝐵) ↔ (𝐴 ∪ (𝐴 +o 𝐵)) = (𝐴 +o 𝐵))
8	6, 7	sylib 217	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ (𝐴 +o 𝐵)) = (𝐴 +o 𝐵))
9	5, 8	eqtrid 2779	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)) = (𝐴 +o 𝐵))
10	4, 9	eqtrd 2767	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} = (𝐴 +o 𝐵))
11		oaun3lem4 42809	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ∈ suc (𝐴 +o 𝐵))
12		unisng 4930	. . . 4 ⊢ ({𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ∈ suc (𝐴 +o 𝐵) → ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)})
13	11, 12	syl 17	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)})
14	10, 13	uneq12d 4163	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}}) = ((𝐴 +o 𝐵) ∪ {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}))
15		uniun 4935	. . 3 ⊢ ∪ ({𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}}) = (∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
16		df-tp 4635	. . . . 5 ⊢ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴), {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}} = ({𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
17		rp-abid 42810	. . . . . . 7 ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}
18	17	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎})
19		oadif1 42812	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)})
20		eqidd 2728	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)})
21	18, 19, 20	tpeq123d 4755	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴), {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}} = {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
22	16, 21	eqtr3id 2781	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}}) = {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
23	22	unieqd 4923	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ ({𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}}) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
24	15, 23	eqtr3id 2781	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}}) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
25		oaun3lem2 42807	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
26		ssequn2 4183	. . 3 ⊢ ({𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵) ↔ ((𝐴 +o 𝐵) ∪ {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}) = (𝐴 +o 𝐵))
27	25, 26	sylib 217	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∪ {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}) = (𝐴 +o 𝐵))
28	14, 24, 27	3eqtr3rd 2776	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2704  ∃wrex 3066  Vcvv 3471   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947  {csn 4630  {cpr 4632  {ctp 4634  ∪ cuni 4910  Oncon0 6372  suc csuc 6374  (class class class)co 7424   +_o coa 8488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-oadd 8495

This theorem is referenced by: (None)

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