![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > onssnum | Structured version Visualization version GIF version |
Description: All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.) |
Ref | Expression |
---|---|
onssnum | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ∈ dom card) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 7746 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
2 | ssorduni 7782 | . . . 4 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
3 | elong 6379 | . . . . 5 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴)) | |
4 | 3 | biimpar 476 | . . . 4 ⊢ ((∪ 𝐴 ∈ V ∧ Ord ∪ 𝐴) → ∪ 𝐴 ∈ On) |
5 | 1, 2, 4 | syl2an 594 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → ∪ 𝐴 ∈ On) |
6 | onsuc 7815 | . . 3 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ On) | |
7 | onenon 9974 | . . 3 ⊢ (suc ∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ dom card) | |
8 | 5, 6, 7 | 3syl 18 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → suc ∪ 𝐴 ∈ dom card) |
9 | onsucuni 7832 | . . 3 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) | |
10 | 9 | adantl 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ⊆ suc ∪ 𝐴) |
11 | ssnum 10064 | . 2 ⊢ ((suc ∪ 𝐴 ∈ dom card ∧ 𝐴 ⊆ suc ∪ 𝐴) → 𝐴 ∈ dom card) | |
12 | 8, 10, 11 | syl2anc 582 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3461 ⊆ wss 3944 ∪ cuni 4909 dom cdm 5678 Ord word 6370 Oncon0 6371 suc csuc 6373 cardccrd 9960 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-er 8725 df-en 8965 df-dom 8966 df-card 9964 |
This theorem is referenced by: dfac12lem3 10170 cfeq0 10281 cfsuc 10282 cff1 10283 cfflb 10284 cflim2 10288 cfss 10290 cfslb 10291 |
Copyright terms: Public domain | W3C validator |