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Mirrors > Home > ILE Home > Th. List > opbrop | Unicode version |
Description: Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.) |
Ref | Expression |
---|---|
opbrop.1 | |
opbrop.2 |
Ref | Expression |
---|---|
opbrop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opbrop.1 | . . . 4 | |
2 | 1 | copsex4g 4219 | . . 3 |
3 | 2 | anbi2d 460 | . 2 |
4 | opexg 4200 | . . 3 | |
5 | opexg 4200 | . . 3 | |
6 | eleq1 2227 | . . . . . 6 | |
7 | 6 | anbi1d 461 | . . . . 5 |
8 | eqeq1 2171 | . . . . . . . 8 | |
9 | 8 | anbi1d 461 | . . . . . . 7 |
10 | 9 | anbi1d 461 | . . . . . 6 |
11 | 10 | 4exbidv 1857 | . . . . 5 |
12 | 7, 11 | anbi12d 465 | . . . 4 |
13 | eleq1 2227 | . . . . . 6 | |
14 | 13 | anbi2d 460 | . . . . 5 |
15 | eqeq1 2171 | . . . . . . . 8 | |
16 | 15 | anbi2d 460 | . . . . . . 7 |
17 | 16 | anbi1d 461 | . . . . . 6 |
18 | 17 | 4exbidv 1857 | . . . . 5 |
19 | 14, 18 | anbi12d 465 | . . . 4 |
20 | opbrop.2 | . . . 4 | |
21 | 12, 19, 20 | brabg 4241 | . . 3 |
22 | 4, 5, 21 | syl2an 287 | . 2 |
23 | opelxpi 4630 | . . . 4 | |
24 | opelxpi 4630 | . . . 4 | |
25 | 23, 24 | anim12i 336 | . . 3 |
26 | 25 | biantrurd 303 | . 2 |
27 | 3, 22, 26 | 3bitr4d 219 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 cvv 2721 cop 3573 class class class wbr 3976 copab 4036 cxp 4596 |
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 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 |
This theorem is referenced by: ecopoveq 6587 oviec 6598 |
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