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| Mirrors > Home > MPE Home > Th. List > ensymd | Structured version Visualization version GIF version | ||
| Description: Symmetry of equinumerosity. Deduction form of ensym 9043. (Contributed by David Moews, 1-May-2017.) |
| Ref | Expression |
|---|---|
| ensymd.1 | ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
| Ref | Expression |
|---|---|
| ensymd | ⊢ (𝜑 → 𝐵 ≈ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ensymd.1 | . 2 ⊢ (𝜑 → 𝐴 ≈ 𝐵) | |
| 2 | ensym 9043 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐵 ≈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 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-10 2141 ax-11 2157 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 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-er 8745 df-en 8986 |
| This theorem is referenced by: f1imaeng 9054 f1imaen2g 9055 xpdom3 9110 omxpen 9114 mapdom2 9188 mapdom3 9189 limensuci 9193 phplem4OLD 9257 phpOLD 9259 unxpdom2 9290 sucxpdom 9291 fiintOLD 9367 marypha1lem 9473 infdifsn 9697 cnfcom2lem 9741 cardidm 9999 cardnueq0 10004 carden2a 10006 card1 10008 cardsdomel 10014 isinffi 10032 en2eqpr 10047 infxpenlem 10053 infxpidm2 10057 alephnbtwn2 10112 alephsucdom 10119 mappwen 10152 finnisoeu 10153 djuen 10210 dju1en 10212 djuassen 10219 xpdjuen 10220 infdju1 10230 pwdju1 10231 onadju 10234 cardadju 10235 djunum 10236 nnadju 10238 ficardadju 10240 ficardun 10241 pwsdompw 10243 infdif2 10249 infxp 10254 ackbij1lem5 10263 cfss 10305 ominf4 10352 isfin4p1 10355 fin23lem27 10368 alephsuc3 10620 canthp1lem1 10692 canthp1lem2 10693 gchdju1 10696 gchinf 10697 pwfseqlem5 10703 pwdjundom 10707 gchdjuidm 10708 gchxpidm 10709 gchhar 10719 inttsk 10814 tskcard 10821 r1tskina 10822 tskuni 10823 hashkf 14371 fz1isolem 14500 isercolllem2 15702 summolem2 15752 zsum 15754 prodmolem2 15971 zprod 15973 4sqlem11 16993 mreexexd 17691 psgnunilem1 19511 simpgnsgd 20120 frlmisfrlm 21868 frlmiscvec 21869 ovoliunlem1 25537 rabfodom 32524 unidifsnel 32553 unidifsnne 32554 fnpreimac 32681 padct 32731 hashpss 32813 lindsdom 37621 matunitlindflem2 37624 heicant 37662 mblfinlem1 37664 sticksstones18 42165 sticksstones19 42166 eldioph2lem1 42771 isnumbasgrplem3 43117 fiuneneq 43204 harval3 43551 enrelmap 44010 enmappw 44012 |
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