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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 8292 | . 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 4888 ≈ cen 8240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-er 8028 df-en 8244 |
This theorem is referenced by: entr2i 8298 entr3i 8299 entr4i 8300 pm54.43 9161 infxpenlem 9171 unsnen 9712 cfpwsdom 9743 tskinf 9928 inar1 9934 gruina 9977 uzenom 13087 znnen 15354 qnnen 15355 rexpen 15370 rucALT 15372 aleph1re 15387 aleph1irr 15388 unben 16028 1stcfb 21668 2ndcredom 21673 hauspwdom 21724 met1stc 22745 ovolctb2 23707 ovolfi 23709 ovoliunlem3 23719 uniiccdif 23793 dyadmbl 23815 mbfimaopnlem 23870 aannenlem3 24533 f1ocnt 30137 dmvlsiga 30798 sigapildsys 30831 omssubadd 30968 carsgclctunlem3 30988 pellex 38373 nnfoctb 40159 nnf1oxpnn 40321 ioonct 40686 caragenunicl 41679 rrx2xpreen 43469 aacllem 43667 |
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