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Mirrors > Home > MPE 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 4197 | . . 3 |
3 | 2 | a1i 11 | . 2 |
4 | opth1.2 | . . . . . . . . 9 | |
5 | 1, 4 | opi1 4719 | . . . . . . . 8 |
6 | id 22 | . . . . . . . 8 | |
7 | 5, 6 | syl5eleq 2551 | . . . . . . 7 |
8 | oprcl 4242 | . . . . . . 7 | |
9 | 7, 8 | syl 16 | . . . . . 6 |
10 | 9 | simpld 459 | . . . . 5 |
11 | prid1g 4136 | . . . . 5 | |
12 | 10, 11 | syl 16 | . . . 4 |
13 | eleq2 2530 | . . . 4 | |
14 | 12, 13 | syl5ibrcom 222 | . . 3 |
15 | elsni 4054 | . . . 4 | |
16 | 15 | eqcomd 2465 | . . 3 |
17 | 14, 16 | syl6 33 | . 2 |
18 | dfopg 4215 | . . . . 5 | |
19 | 7, 8, 18 | 3syl 20 | . . . 4 |
20 | 7, 19 | eleqtrd 2547 | . . 3 |
21 | elpri 4049 | . . 3 | |
22 | 20, 21 | syl 16 | . 2 |
23 | 3, 17, 22 | mpjaod 381 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 \/ wo 368
/\ wa 369 = wceq 1395 e. wcel 1818
cvv 3109
{ csn 4029 { cpr 4031 <. cop 4035 |
This theorem is referenced by: opth 4726 dmsnopg 5484 funcnvsn 5638 oprabid 6323 seqomlem2 7135 unxpdomlem3 7746 dfac5lem4 8528 dcomex 8848 canthwelem 9049 uzrdgfni 12069 gsum2d2 17002 2trllemA 24552 2pthon 24604 2pthon3v 24606 constr3lem2 24646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pr 4691 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-v 3111 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-sn 4030 df-pr 4032 df-op 4036 |
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