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Mirrors > Home > ILE Home > Th. List > elcnv2 | GIF version |
Description: Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.) |
Ref | Expression |
---|---|
elcnv2 | ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑦, 𝑥〉 ∈ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elcnv 4822 | . 2 ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥)) | |
2 | df-br 4019 | . . . 4 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑦, 𝑥〉 ∈ 𝑅) | |
3 | 2 | anbi2i 457 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥) ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑦, 𝑥〉 ∈ 𝑅)) |
4 | 3 | 2exbii 1617 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑦, 𝑥〉 ∈ 𝑅)) |
5 | 1, 4 | bitri 184 | 1 ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑦, 𝑥〉 ∈ 𝑅)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1503 ∈ wcel 2160 〈cop 3610 class class class wbr 4018 ◡ccnv 4643 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-cnv 4652 |
This theorem is referenced by: cnvuni 4831 |
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