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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 7873 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ≈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-en 7842 |
This theorem is referenced by: ener 7888 enerOLD 7889 en0 7905 pwen 8018 phplem2 8025 phplem3 8026 isinf 8058 pssnn 8063 karden 8641 mappwen 8818 cdacomen 8886 infmap2 8923 ackbij1lem5 8929 axcc4dom 9146 domtriomlem 9147 cfpwsdom 9285 0tsk 9456 fzennn 12629 qnnen 14781 rpnnen 14795 rexpen 14796 lmisfree 20000 met2ndci 22137 lgseisenlem2 24901 poimirlem9 32588 poimirlem26 32605 |
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