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Mirrors > Home > ILE Home > Th. List > opthg | Unicode version |
Description: Ordered pair theorem. and are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opthg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3752 | . . . 4 | |
2 | 1 | eqeq1d 2173 | . . 3 |
3 | eqeq1 2171 | . . . 4 | |
4 | 3 | anbi1d 461 | . . 3 |
5 | 2, 4 | bibi12d 234 | . 2 |
6 | opeq2 3753 | . . . 4 | |
7 | 6 | eqeq1d 2173 | . . 3 |
8 | eqeq1 2171 | . . . 4 | |
9 | 8 | anbi2d 460 | . . 3 |
10 | 7, 9 | bibi12d 234 | . 2 |
11 | vex 2724 | . . 3 | |
12 | vex 2724 | . . 3 | |
13 | 11, 12 | opth 4209 | . 2 |
14 | 5, 10, 13 | vtocl2g 2785 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 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: opthg2 4211 xpopth 6136 eqop 6137 inl11 7021 preqlu 7404 cauappcvgprlemladd 7590 elrealeu 7761 qnumdenbi 12101 crth 12133 |
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