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Mirrors > Home > MPE Home > Th. List > ensymi | Structured version Visualization version GIF version |
Description: Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.) |
Ref | Expression |
---|---|
ensymi.2 | ⊢ 𝐴 ≈ 𝐵 |
Ref | Expression |
---|---|
ensymi | ⊢ 𝐵 ≈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensymi.2 | . 2 ⊢ 𝐴 ≈ 𝐵 | |
2 | ensym 7891 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐵 ≈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: 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-er 7629 df-en 7842 |
This theorem is referenced by: entr2i 7897 entr3i 7898 entr4i 7899 pm54.43 8709 infxpenlem 8719 ackbij1lem5 8929 unsnen 9254 cfpwsdom 9285 tskinf 9470 inar1 9476 gruina 9519 uzenom 12625 znnen 14780 qnnen 14781 rexpen 14796 rucALT 14798 aleph1re 14813 aleph1irr 14814 unben 15451 1stcfb 21058 2ndcredom 21063 hauspwdom 21114 met1stc 22136 ovolctb2 23067 ovolfi 23069 ovoliunlem3 23079 uniiccdif 23152 dyadmbl 23174 mbfimaopnlem 23228 aannenlem3 23889 f1ocnt 28946 dmvlsiga 29519 sigapildsys 29552 omssubadd 29689 carsgclctunlem3 29709 pellex 36417 nnfoctb 38238 nnf1oxpnn 38379 ioonct 38611 caragenunicl 39414 aacllem 42356 |
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