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Mirrors > Home > ILE Home > Th. List > ensymi | 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 6774 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐵 ≈ 𝐴 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4000 ≈ cen 6731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-er 6528 df-en 6734 |
This theorem is referenced by: entr2i 6780 entr3i 6781 entr4i 6782 omp1eom 7087 pm54.43 7182 dju1p1e2 7189 pw1dom2 7219 1nprm 12084 unennn 12368 ennnfonelemen 12392 ennnfonelemim 12395 exmidunben 12397 qnnen 12402 ctiunct 12411 nninfdc 12424 iooreen 14406 |
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