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Mirrors > Home > ILE Home > Th. List > op1steq | Unicode version |
Description: Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
op1steq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss 4446 | . . 3 | |
2 | 1 | sseli 2941 | . 2 |
3 | eqid 2040 | . . . . . 6 | |
4 | eqopi 5798 | . . . . . 6 | |
5 | 3, 4 | mpanr2 414 | . . . . 5 |
6 | 2ndexg 5795 | . . . . . . 7 | |
7 | opeq2 3550 | . . . . . . . . 9 | |
8 | 7 | eqeq2d 2051 | . . . . . . . 8 |
9 | 8 | spcegv 2641 | . . . . . . 7 |
10 | 6, 9 | syl 14 | . . . . . 6 |
11 | 10 | adantr 261 | . . . . 5 |
12 | 5, 11 | mpd 13 | . . . 4 |
13 | 12 | ex 108 | . . 3 |
14 | eqop 5803 | . . . . 5 | |
15 | simpl 102 | . . . . 5 | |
16 | 14, 15 | syl6bi 152 | . . . 4 |
17 | 16 | exlimdv 1700 | . . 3 |
18 | 13, 17 | impbid 120 | . 2 |
19 | 2, 18 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 cop 3378 cxp 4343 cfv 4902 c1st 5765 c2nd 5766 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fo 4908 df-fv 4910 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: releldm2 5811 |
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