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| Mirrors > Home > MPE Home > Th. List > isnumi | Structured version Visualization version GIF version | ||
| Description: A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
| Ref | Expression |
|---|---|
| isnumi | ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | breq1 5146 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵)) | |
| 2 | 1 | rspcev 3622 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → ∃𝑥 ∈ On 𝑥 ≈ 𝐵) |
| 3 | isnum2 9985 | . 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐵) | |
| 4 | 2, 3 | sylibr 234 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 dom cdm 5685 Oncon0 6384 ≈ cen 8982 cardccrd 9975 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-fun 6563 df-fn 6564 df-f 6565 df-en 8986 df-card 9979 |
| This theorem is referenced by: finnum 9988 onenon 9989 tskwe 9990 xpnum 9991 isnum3 9994 dfac8alem 10069 djunum 10236 fin67 10435 isfin7-2 10436 gch2 10715 gchacg 10720 znnen 16248 qnnen 16249 met1stc 24534 re2ndc 24822 uniiccdif 25613 dyadmbl 25635 opnmblALT 25638 mbfimaopnlem 25690 aannenlem3 26372 poimirlem32 37659 |
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