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| Description: Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| enrefg | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | f1oi 6886 | . . 3 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
| 2 | f1oen2g 9009 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴) → 𝐴 ≈ 𝐴) | |
| 3 | 1, 2 | mp3an3 1452 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ 𝐴) | 
| 4 | 3 | anidms 566 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 I cid 5577 ↾ cres 5687 –1-1-onto→wf1o 6560 ≈ cen 8982 | 
| 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-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 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-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 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-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-en 8986 | 
| This theorem is referenced by: enref 9025 eqeng 9026 domrefg 9027 difsnen 9093 sdomirr 9154 mapdom1 9182 mapdom2 9188 rneqdmfinf1o 9373 infdifsn 9697 infdiffi 9698 onenon 9989 cardonle 9997 dju1en 10212 xpdjuen 10220 mapdjuen 10221 onadju 10234 nnadju 10238 ssfin4 10350 canthp1lem1 10692 gchhar 10719 hashfac 14497 mreexexlem3d 17689 cyggenod 19902 mdetunilem8 22625 frlmpwfi 43110 fiuneneq 43204 enrelmap 44010 | 
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