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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 9024 | . 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 5149 ≈ cen 8961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-er 8725 df-en 8965 |
This theorem is referenced by: entr2i 9030 entr3i 9031 entr4i 9032 pm54.43 10026 infxpenlem 10038 unsnen 10578 cfpwsdom 10609 tskinf 10794 inar1 10800 gruina 10843 uzenom 13965 znnen 16192 qnnen 16193 rexpen 16208 rucALT 16210 aleph1re 16225 aleph1irr 16226 unben 16881 1stcfb 23393 2ndcredom 23398 hauspwdom 23449 met1stc 24474 ovolctb2 25465 ovolfi 25467 ovoliunlem3 25477 uniiccdif 25551 dyadmbl 25573 mbfimaopnlem 25628 aannenlem3 26310 f1ocnt 32652 dmvlsiga 33879 sigapildsys 33912 omssubadd 34051 carsgclctunlem3 34071 pellex 42397 tr3dom 43100 nnfoctb 44553 nnf1oxpnn 44707 ioonct 45060 caragenunicl 46050 rrx2xpreen 47978 aacllem 48420 |
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