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Theorem cplem2 9919

Description: Lemma for the Collection Principle cp 9920. (Contributed by NM, 17-Oct-2003.)

**Hypothesis**
Ref	Expression
cplem2.1	⊢ 𝐴 ∈ V

**Assertion**
Ref	Expression
cplem2	⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑦,𝐵 Allowed substitution hint: 𝐵(𝑥)

Proof of Theorem cplem2 Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cplem2.1	. . 3 ⊢ 𝐴 ∈ V
2		scottex 9914	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
3	1, 2	iunex 7976	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
4		nfiu1 5032	. . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
5	4	nfeq2 2916	. . 3 ⊢ Ⅎ𝑥 𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
6		ineq2 4206	. . . . 5 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (𝐵 ∩ 𝑦) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}))
7	6	neeq1d 2996	. . . 4 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵 ∩ 𝑦) ≠ ∅ ↔ (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅))
8	7	imbi2d 339	. . 3 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅) ↔ (𝐵 ≠ ∅ → (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
9	5, 8	ralbid 3266	. 2 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅) ↔ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
10		eqid 2727	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
11		eqid 2727	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
12	10, 11	cplem1 9918	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)
13	3, 9, 12	ceqsexv2d 3526	1 ⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2936 ∀wral 3057 {crab 3428 Vcvv 3471 ∩ cin 3946 ⊆ wss 3947 ∅c0 4324 ∪ ciun 4998 ‘cfv 6551 rankcrnk 9792

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-pow 5367 ax-pr 5431 ax-un 7744 ax-reg 9621 ax-inf2 9670

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-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-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 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-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-r1 9793 df-rank 9794

This theorem is referenced by: cp 9920

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