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 6853 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
2 | ssorduni 6877 | . . . 4 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
3 | elong 5648 | . . . . 5 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴)) | |
4 | 3 | biimpar 501 | . . . 4 ⊢ ((∪ 𝐴 ∈ V ∧ Ord ∪ 𝐴) → ∪ 𝐴 ∈ On) |
5 | 1, 2, 4 | syl2an 493 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → ∪ 𝐴 ∈ On) |
6 | suceloni 6905 | . . 3 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ On) | |
7 | onenon 8658 | . . 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 6920 | . . 3 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) | |
10 | 9 | adantl 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ⊆ suc ∪ 𝐴) |
11 | ssnum 8745 | . 2 ⊢ ((suc ∪ 𝐴 ∈ dom card ∧ 𝐴 ⊆ suc ∪ 𝐴) → 𝐴 ∈ dom card) | |
12 | 8, 10, 11 | syl2anc 691 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∪ cuni 4372 dom cdm 5038 Ord word 5639 Oncon0 5640 suc csuc 5642 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-wrecs 7294 df-recs 7355 df-er 7629 df-en 7842 df-dom 7843 df-card 8648 |
This theorem is referenced by: dfac12lem3 8850 cfeq0 8961 cfsuc 8962 cff1 8963 cfflb 8964 cflim2 8968 cfss 8970 cfslb 8971 |
Copyright terms: Public domain | W3C validator |