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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 5042 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵)) | |
2 | 1 | rspcev 3527 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → ∃𝑥 ∈ On 𝑥 ≈ 𝐵) |
3 | isnum2 9526 | . 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐵) | |
4 | 2, 3 | sylibr 237 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2112 ∃wrex 3052 class class class wbr 5039 dom cdm 5536 Oncon0 6191 ≈ cen 8601 cardccrd 9516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-fun 6360 df-fn 6361 df-f 6362 df-en 8605 df-card 9520 |
This theorem is referenced by: finnum 9529 onenon 9530 tskwe 9531 xpnum 9532 isnum3 9535 dfac8alem 9608 djunum 9774 fin67 9974 isfin7-2 9975 gch2 10254 gchacg 10259 znnen 15736 qnnen 15737 met1stc 23373 re2ndc 23652 uniiccdif 24429 dyadmbl 24451 opnmblALT 24454 mbfimaopnlem 24506 aannenlem3 25177 poimirlem32 35495 |
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