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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 7468 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
2 | ssorduni 7502 | . . . 4 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
3 | elong 6201 | . . . . 5 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴)) | |
4 | 3 | biimpar 480 | . . . 4 ⊢ ((∪ 𝐴 ∈ V ∧ Ord ∪ 𝐴) → ∪ 𝐴 ∈ On) |
5 | 1, 2, 4 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → ∪ 𝐴 ∈ On) |
6 | suceloni 7530 | . . 3 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ On) | |
7 | onenon 9380 | . . 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 7545 | . . 3 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) | |
10 | 9 | adantl 484 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ⊆ suc ∪ 𝐴) |
11 | ssnum 9467 | . 2 ⊢ ((suc ∪ 𝐴 ∈ dom card ∧ 𝐴 ⊆ suc ∪ 𝐴) → 𝐴 ∈ dom card) | |
12 | 8, 10, 11 | syl2anc 586 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ∪ cuni 4840 dom cdm 5557 Ord word 6192 Oncon0 6193 suc csuc 6195 cardccrd 9366 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-wrecs 7949 df-recs 8010 df-er 8291 df-en 8512 df-dom 8513 df-card 9370 |
This theorem is referenced by: dfac12lem3 9573 cfeq0 9680 cfsuc 9681 cff1 9682 cfflb 9683 cflim2 9687 cfss 9689 cfslb 9690 |
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