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Mirrors > Home > MPE Home > Th. List > ensym | Structured version Visualization version GIF version |
Description: Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
ensym | ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensymb 8169 | . 2 ⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴) | |
2 | 1 | biimpi 206 | 1 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 4804 ≈ cen 8118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-er 7911 df-en 8122 |
This theorem is referenced by: ensymi 8171 ensymd 8172 sbthb 8246 domnsym 8251 sdomdomtr 8258 domsdomtr 8260 enen1 8265 enen2 8266 domen1 8267 domen2 8268 sdomen1 8269 sdomen2 8270 domtriord 8271 xpen 8288 pwen 8298 nneneq 8308 php2 8310 php3 8311 ominf 8337 fineqvlem 8339 en1eqsn 8355 dif1en 8358 enp1i 8360 findcard3 8368 isfinite2 8383 nnsdomg 8384 domunfican 8398 infcntss 8399 fiint 8402 wdomen1 8646 wdomen2 8647 unxpwdom2 8658 karden 8931 finnum 8964 carden2b 8983 fidomtri2 9010 cardmin2 9014 pr2ne 9018 en2eleq 9021 infxpenlem 9026 acnen 9066 acnen2 9068 infpwfien 9075 alephordi 9087 alephinit 9108 dfac12lem2 9158 dfac12r 9160 uncdadom 9185 cdacomen 9195 cdainf 9206 pwsdompw 9218 infmap2 9232 ackbij1b 9253 cflim2 9277 fin4en1 9323 domfin4 9325 fin23lem25 9338 fin23lem23 9340 enfin1ai 9398 fin67 9409 isfin7-2 9410 fin1a2lem11 9424 axcc2lem 9450 axcclem 9471 numthcor 9508 carden 9565 sdomsdomcard 9574 canthnum 9663 canthwe 9665 canthp1lem2 9667 canthp1 9668 pwxpndom2 9679 gchcdaidm 9682 gchxpidm 9683 gchpwdom 9684 inawinalem 9703 grudomon 9831 isfinite4 13345 hashfn 13356 ramub2 15920 dfod2 18181 sylow2blem1 18235 znhash 20109 hauspwdom 21506 rectbntr0 22836 ovolctb 23458 dyadmbl 23568 eupthfi 27357 derangen 31461 finminlem 32618 phpreu 33706 pellexlem4 37898 pellexlem5 37899 pellex 37901 |
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