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Mirrors > Home > MPE Home > Th. List > enrefg | Structured version Visualization version GIF version |
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 6858 | . . 3 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
2 | f1oen2g 8947 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴) → 𝐴 ≈ 𝐴) | |
3 | 1, 2 | mp3an3 1450 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ 𝐴) |
4 | 3 | anidms 567 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5141 I cid 5566 ↾ cres 5671 –1-1-onto→wf1o 6531 ≈ cen 8919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-en 8923 |
This theorem is referenced by: enref 8964 eqeng 8965 domrefg 8966 difsnen 9036 sdomirr 9097 mapdom1 9125 mapdom2 9131 rneqdmfinf1o 9311 infdifsn 9634 infdiffi 9635 onenon 9926 cardonle 9934 dju1en 10148 xpdjuen 10156 mapdjuen 10157 onadju 10170 nnadju 10174 ssfin4 10287 canthp1lem1 10629 gchhar 10656 hashfac 14401 mreexexlem3d 17572 cyggenod 19711 fidomndrnglem 20859 mdetunilem8 22050 frlmpwfi 41611 fiuneneq 41710 enrelmap 42519 |
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