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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 6652 | . . 3 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
2 | f1oen2g 8526 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴) → 𝐴 ≈ 𝐴) | |
3 | 1, 2 | mp3an3 1446 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ 𝐴) |
4 | 3 | anidms 569 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5066 I cid 5459 ↾ cres 5557 –1-1-onto→wf1o 6354 ≈ cen 8506 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-en 8510 |
This theorem is referenced by: enref 8542 eqeng 8543 domrefg 8544 difsnen 8599 sdomirr 8654 mapdom1 8682 mapdom2 8688 onfin 8709 ssnnfi 8737 rneqdmfinf1o 8800 infdifsn 9120 infdiffi 9121 onenon 9378 cardonle 9386 dju1en 9597 xpdjuen 9605 mapdjuen 9606 onadju 9619 ssfin4 9732 canthp1lem1 10074 gchhar 10101 hashfac 13817 mreexexlem3d 16917 cyggenod 19003 fidomndrnglem 20079 mdetunilem8 21228 frlmpwfi 39718 fiuneneq 39817 enrelmap 40363 |
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