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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 7734 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
2 | ssorduni 7770 | . . . 4 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
3 | elong 6372 | . . . . 5 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴)) | |
4 | 3 | biimpar 477 | . . . 4 ⊢ ((∪ 𝐴 ∈ V ∧ Ord ∪ 𝐴) → ∪ 𝐴 ∈ On) |
5 | 1, 2, 4 | syl2an 595 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → ∪ 𝐴 ∈ On) |
6 | onsuc 7803 | . . 3 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ On) | |
7 | onenon 9950 | . . 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 7820 | . . 3 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) | |
10 | 9 | adantl 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ⊆ suc ∪ 𝐴) |
11 | ssnum 10040 | . 2 ⊢ ((suc ∪ 𝐴 ∈ dom card ∧ 𝐴 ⊆ suc ∪ 𝐴) → 𝐴 ∈ dom card) | |
12 | 8, 10, 11 | syl2anc 583 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3948 ∪ cuni 4908 dom cdm 5676 Ord word 6363 Oncon0 6364 suc csuc 6366 cardccrd 9936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-er 8709 df-en 8946 df-dom 8947 df-card 9940 |
This theorem is referenced by: dfac12lem3 10146 cfeq0 10257 cfsuc 10258 cff1 10259 cfflb 10260 cflim2 10264 cfss 10266 cfslb 10267 |
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