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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 6755 | . 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 3987 ≈ cen 6713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-er 6510 df-en 6716 |
This theorem is referenced by: ensymi 6757 ensymd 6758 enen1 6815 enen2 6816 domen1 6817 domen2 6818 nneneq 6832 ssfilem 6850 diffitest 6862 fiintim 6903 fisseneq 6906 en1eqsn 6922 fidcenumlemim 6926 enomni 7112 enmkv 7135 enwomni 7143 finnum 7149 pr2ne 7158 djucomen 7182 cc2lem 7217 enct 12377 |
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