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Mirrors > Home > ILE Home > Th. List > opeqpr | Unicode version |
Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) |
Ref | Expression |
---|---|
opeqpr.1 | |
opeqpr.2 | |
opeqpr.3 | |
opeqpr.4 |
Ref | Expression |
---|---|
opeqpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2141 | . 2 | |
2 | opeqpr.1 | . . . 4 | |
3 | opeqpr.2 | . . . 4 | |
4 | 2, 3 | dfop 3704 | . . 3 |
5 | 4 | eqeq2i 2150 | . 2 |
6 | opeqpr.3 | . . 3 | |
7 | opeqpr.4 | . . 3 | |
8 | 2 | snex 4109 | . . 3 |
9 | prexg 4133 | . . . 4 | |
10 | 2, 3, 9 | mp2an 422 | . . 3 |
11 | 6, 7, 8, 10 | preq12b 3697 | . 2 |
12 | 1, 5, 11 | 3bitri 205 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wo 697 wceq 1331 wcel 1480 cvv 2686 csn 3527 cpr 3528 cop 3530 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 |
This theorem is referenced by: relop 4689 |
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