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Mirrors > Home > ILE Home > Th. List > ensym | 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 6626 | . 2 ⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴) | |
2 | 1 | biimpi 119 | 1 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 class class class wbr 3893 ≈ cen 6584 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-er 6381 df-en 6587 |
This theorem is referenced by: ensymi 6628 ensymd 6629 enen1 6685 enen2 6686 domen1 6687 domen2 6688 nneneq 6702 ssfilem 6720 diffitest 6732 fiintim 6768 fisseneq 6771 en1eqsn 6786 fidcenumlemim 6790 enomni 6959 finnum 6986 pr2ne 6995 djucomen 7017 |
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