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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 6174 | . . 3 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
2 | f1oen2g 7972 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴) → 𝐴 ≈ 𝐴) | |
3 | 1, 2 | mp3an3 1413 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ 𝐴) |
4 | 3 | anidms 677 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 class class class wbr 4653 I cid 5023 ↾ cres 5116 –1-1-onto→wf1o 5887 ≈ cen 7952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-en 7956 |
This theorem is referenced by: enref 7988 eqeng 7989 domrefg 7990 difsnen 8042 sdomirr 8097 mapdom1 8125 mapdom2 8131 onfin 8151 ssnnfi 8179 rneqdmfinf1o 8242 infdifsn 8554 infdiffi 8555 onenon 8775 cardonle 8783 cda1en 8997 xpcdaen 9005 mapcdaen 9006 onacda 9019 ssfin4 9132 canthp1lem1 9474 gchhar 9501 hashfac 13242 mreexexlem3d 16306 cyggenod 18286 fidomndrnglem 19306 mdetunilem8 20425 frlmpwfi 37668 fiuneneq 37775 enrelmap 38291 |
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