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Mirrors > Home > ILE Home > Th. List > cnvsn | GIF version |
Description: Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
cnvsn.1 | ⊢ 𝐴 ∈ V |
cnvsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
cnvsn | ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnvsn 4985 | . 2 ⊢ ◡◡{〈𝐵, 𝐴〉} = ◡{〈𝐴, 𝐵〉} | |
2 | cnvsn.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | cnvsn.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | 2, 3 | relsnop 4615 | . . 3 ⊢ Rel {〈𝐵, 𝐴〉} |
5 | dfrel2 4959 | . . 3 ⊢ (Rel {〈𝐵, 𝐴〉} ↔ ◡◡{〈𝐵, 𝐴〉} = {〈𝐵, 𝐴〉}) | |
6 | 4, 5 | mpbi 144 | . 2 ⊢ ◡◡{〈𝐵, 𝐴〉} = {〈𝐵, 𝐴〉} |
7 | 1, 6 | eqtr3i 2140 | 1 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∈ wcel 1465 Vcvv 2660 {csn 3497 〈cop 3500 ◡ccnv 4508 Rel wrel 4514 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 |
This theorem is referenced by: op2ndb 4992 cnvsng 4994 f1osn 5375 |
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