![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elcnv2 | Structured version Visualization version GIF version |
Description: Membership in a converse relation. 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 5879 | . 2 ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥)) | |
2 | df-br 5150 | . . . 4 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑦, 𝑥〉 ∈ 𝑅) | |
3 | 2 | anbi2i 621 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥) ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑦, 𝑥〉 ∈ 𝑅)) |
4 | 3 | 2exbii 1843 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑦, 𝑥〉 ∈ 𝑅)) |
5 | 1, 4 | bitri 274 | 1 ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑦, 𝑥〉 ∈ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 〈cop 4636 class class class wbr 5149 ◡ccnv 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-cnv 5686 |
This theorem is referenced by: cnvuni 5889 elcnvlem 43173 |
Copyright terms: Public domain | W3C validator |