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Mirrors > Home > ILE Home > Th. List > opth1 | Unicode version |
Description: Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opth1.1 | |
opth1.2 |
Ref | Expression |
---|---|
opth1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth1.1 | . . . 4 | |
2 | 1 | sneqr 3734 | . . 3 |
3 | 2 | a1i 9 | . 2 |
4 | opth1.2 | . . . . . . . . 9 | |
5 | 1, 4 | opi1 4204 | . . . . . . . 8 |
6 | id 19 | . . . . . . . 8 | |
7 | 5, 6 | eleqtrid 2253 | . . . . . . 7 |
8 | oprcl 3776 | . . . . . . 7 | |
9 | 7, 8 | syl 14 | . . . . . 6 |
10 | 9 | simpld 111 | . . . . 5 |
11 | prid1g 3674 | . . . . 5 | |
12 | 10, 11 | syl 14 | . . . 4 |
13 | eleq2 2228 | . . . 4 | |
14 | 12, 13 | syl5ibrcom 156 | . . 3 |
15 | elsni 3588 | . . . 4 | |
16 | 15 | eqcomd 2170 | . . 3 |
17 | 14, 16 | syl6 33 | . 2 |
18 | dfopg 3750 | . . . . 5 | |
19 | 7, 8, 18 | 3syl 17 | . . . 4 |
20 | 7, 19 | eleqtrd 2243 | . . 3 |
21 | elpri 3593 | . . 3 | |
22 | 20, 21 | syl 14 | . 2 |
23 | 3, 17, 22 | mpjaod 708 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 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 |
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: opth 4209 dmsnopg 5069 funcnvsn 5227 oprabid 5865 pwle2 13719 |
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