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| Mirrors > Home > MPE Home > Th. List > enref | Structured version Visualization version GIF version | ||
| Description: Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.) |
| Ref | Expression |
|---|---|
| enref.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| enref | ⊢ 𝐴 ≈ 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | enref.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | enrefg 9024 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ≈ 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ≈ 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: ener 9041 en0ALT 9059 pwen 9190 phplem2OLD 9255 phplem3OLD 9256 isinfOLD 9297 karden 9935 mappwen 10152 nnadju 10238 infmap2 10257 ackbij1lem5 10263 axcc4dom 10481 domtriomlem 10482 cfpwsdom 10624 0tsk 10795 fzennn 14009 qnnen 16249 rpnnen 16263 rexpen 16264 lmisfree 21862 met2ndci 24535 lgseisenlem2 27420 poimirlem9 37636 poimirlem26 37653 1aryenef 48566 2aryenef 48577 |
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