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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 3705 | . . . 4 | |
2 | 1 | eqeq1d 2148 | . . 3 |
3 | eqeq1 2146 | . . . 4 | |
4 | 3 | anbi1d 460 | . . 3 |
5 | 2, 4 | bibi12d 234 | . 2 |
6 | opeq2 3706 | . . . 4 | |
7 | 6 | eqeq1d 2148 | . . 3 |
8 | eqeq1 2146 | . . . 4 | |
9 | 8 | anbi2d 459 | . . 3 |
10 | 7, 9 | bibi12d 234 | . 2 |
11 | vex 2689 | . . 3 | |
12 | vex 2689 | . . 3 | |
13 | 11, 12 | opth 4159 | . 2 |
14 | 5, 10, 13 | vtocl2g 2750 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 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: opthg2 4161 xpopth 6074 eqop 6075 inl11 6950 preqlu 7280 cauappcvgprlemladd 7466 elrealeu 7637 qnumdenbi 11870 crth 11900 |
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