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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 4719 | . . 3 | |
2 | 1 | sseli 3143 | . 2 |
3 | eqid 2170 | . . . . . 6 | |
4 | eqopi 6151 | . . . . . 6 | |
5 | 3, 4 | mpanr2 436 | . . . . 5 |
6 | 2ndexg 6147 | . . . . . . 7 | |
7 | opeq2 3766 | . . . . . . . . 9 | |
8 | 7 | eqeq2d 2182 | . . . . . . . 8 |
9 | 8 | spcegv 2818 | . . . . . . 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 6156 | . . . . 5 | |
15 | simpl 108 | . . . . 5 | |
16 | 14, 15 | syl6bi 162 | . . . 4 |
17 | 16 | exlimdv 1812 | . . 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 1348 wex 1485 wcel 2141 cvv 2730 cop 3586 cxp 4609 cfv 5198 c1st 6117 c2nd 6118 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fo 5204 df-fv 5206 df-1st 6119 df-2nd 6120 |
This theorem is referenced by: releldm2 6164 |
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