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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 5141 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵)) | |
2 | 1 | rspcev 3604 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → ∃𝑥 ∈ On 𝑥 ≈ 𝐵) |
3 | isnum2 9936 | . 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐵) | |
4 | 2, 3 | sylibr 233 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ∃wrex 3062 class class class wbr 5138 dom cdm 5666 Oncon0 6354 ≈ cen 8932 cardccrd 9926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6357 df-on 6358 df-fun 6535 df-fn 6536 df-f 6537 df-en 8936 df-card 9930 |
This theorem is referenced by: finnum 9939 onenon 9940 tskwe 9941 xpnum 9942 isnum3 9945 dfac8alem 10020 djunum 10186 fin67 10386 isfin7-2 10387 gch2 10666 gchacg 10671 znnen 16152 qnnen 16153 met1stc 24352 re2ndc 24639 uniiccdif 25429 dyadmbl 25451 opnmblALT 25454 mbfimaopnlem 25506 aannenlem3 26184 poimirlem32 37010 |
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