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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 4642 | . . 3 | |
2 | 1 | sseli 3088 | . 2 |
3 | eqid 2137 | . . . . . 6 | |
4 | eqopi 6063 | . . . . . 6 | |
5 | 3, 4 | mpanr2 434 | . . . . 5 |
6 | 2ndexg 6059 | . . . . . . 7 | |
7 | opeq2 3701 | . . . . . . . . 9 | |
8 | 7 | eqeq2d 2149 | . . . . . . . 8 |
9 | 8 | spcegv 2769 | . . . . . . 7 |
10 | 6, 9 | syl 14 | . . . . . 6 |
11 | 10 | adantr 274 | . . . . 5 |
12 | 5, 11 | mpd 13 | . . . 4 |
13 | 12 | ex 114 | . . 3 |
14 | eqop 6068 | . . . . 5 | |
15 | simpl 108 | . . . . 5 | |
16 | 14, 15 | syl6bi 162 | . . . 4 |
17 | 16 | exlimdv 1791 | . . 3 |
18 | 13, 17 | impbid 128 | . 2 |
19 | 2, 18 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cvv 2681 cop 3525 cxp 4532 cfv 5118 c1st 6029 c2nd 6030 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fo 5124 df-fv 5126 df-1st 6031 df-2nd 6032 |
This theorem is referenced by: releldm2 6076 |
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