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Mirrors > Home > MPE Home > Th. List > ennum | Structured version Visualization version GIF version |
Description: Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
ennum | ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enen2 9124 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵)) | |
2 | 1 | rexbidv 3177 | . 2 ⊢ (𝐴 ≈ 𝐵 → (∃𝑥 ∈ On 𝑥 ≈ 𝐴 ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐵)) |
3 | isnum2 9946 | . 2 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) | |
4 | isnum2 9946 | . 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐵) | |
5 | 2, 3, 4 | 3bitr4g 314 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2105 ∃wrex 3069 class class class wbr 5148 dom cdm 5676 Oncon0 6364 ≈ cen 8942 cardccrd 9936 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-er 8709 df-en 8946 df-card 9940 |
This theorem is referenced by: carden2b 9968 dfac12lem3 10146 dfac12k 10148 qnnen 16163 cygctb 19808 |
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