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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 2166 | . 2 | |
2 | opeqpr.1 | . . . 4 | |
3 | opeqpr.2 | . . . 4 | |
4 | 2, 3 | dfop 3751 | . . 3 |
5 | 4 | eqeq2i 2175 | . 2 |
6 | opeqpr.3 | . . 3 | |
7 | opeqpr.4 | . . 3 | |
8 | 2 | snex 4158 | . . 3 |
9 | prexg 4183 | . . . 4 | |
10 | 2, 3, 9 | mp2an 423 | . . 3 |
11 | 6, 7, 8, 10 | preq12b 3744 | . 2 |
12 | 1, 5, 11 | 3bitri 205 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wo 698 wceq 1342 wcel 2135 cvv 2721 csn 3570 cpr 3571 cop 3573 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 |
This theorem is referenced by: relop 4748 |
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