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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 4876 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵)) | |
2 | 1 | rspcev 3526 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → ∃𝑥 ∈ On 𝑥 ≈ 𝐵) |
3 | isnum2 9084 | . 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐵) | |
4 | 2, 3 | sylibr 226 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 ∃wrex 3118 class class class wbr 4873 dom cdm 5342 Oncon0 5963 ≈ cen 8219 cardccrd 9074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-fun 6125 df-fn 6126 df-f 6127 df-en 8223 df-card 9078 |
This theorem is referenced by: finnum 9087 onenon 9088 tskwe 9089 xpnum 9090 isnum3 9093 dfac8alem 9165 cdanum 9336 fin67 9532 isfin7-2 9533 gch2 9812 gchacg 9817 znnen 15315 qnnen 15316 met1stc 22696 re2ndc 22974 uniiccdif 23744 dyadmbl 23766 opnmblALT 23769 mbfimaopnlem 23821 aannenlem3 24484 poimirlem32 33985 |
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