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Mirrors > Home > ILE Home > Th. List > elcnv | GIF version |
Description: Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.) |
Ref | Expression |
---|---|
elcnv | ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnv 4371 | . . 3 ⊢ ◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑦𝑅𝑥} | |
2 | 1 | eleq2i 2145 | . 2 ⊢ (𝐴 ∈ ◡𝑅 ↔ 𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑦𝑅𝑥}) |
3 | elopab 4013 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑦𝑅𝑥} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥)) | |
4 | 2, 3 | bitri 182 | 1 ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 〈cop 3401 class class class wbr 3785 {copab 3838 ◡ccnv 4362 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-cnv 4371 |
This theorem is referenced by: elcnv2 4531 |
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