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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 3984 | . . 3 |
3 | 2 | anbi2d 437 | . 2 |
4 | elex 2566 | . . . 4 | |
5 | elex 2566 | . . . 4 | |
6 | opexgOLD 3965 | . . . 4 | |
7 | 4, 5, 6 | syl2an 273 | . . 3 |
8 | elex 2566 | . . . 4 | |
9 | elex 2566 | . . . 4 | |
10 | opexgOLD 3965 | . . . 4 | |
11 | 8, 9, 10 | syl2an 273 | . . 3 |
12 | eleq1 2100 | . . . . . 6 | |
13 | 12 | anbi1d 438 | . . . . 5 |
14 | eqeq1 2046 | . . . . . . . 8 | |
15 | 14 | anbi1d 438 | . . . . . . 7 |
16 | 15 | anbi1d 438 | . . . . . 6 |
17 | 16 | 4exbidv 1750 | . . . . 5 |
18 | 13, 17 | anbi12d 442 | . . . 4 |
19 | eleq1 2100 | . . . . . 6 | |
20 | 19 | anbi2d 437 | . . . . 5 |
21 | eqeq1 2046 | . . . . . . . 8 | |
22 | 21 | anbi2d 437 | . . . . . . 7 |
23 | 22 | anbi1d 438 | . . . . . 6 |
24 | 23 | 4exbidv 1750 | . . . . 5 |
25 | 20, 24 | anbi12d 442 | . . . 4 |
26 | opbrop.2 | . . . 4 | |
27 | 18, 25, 26 | brabg 4006 | . . 3 |
28 | 7, 11, 27 | syl2an 273 | . 2 |
29 | opelxpi 4376 | . . . 4 | |
30 | opelxpi 4376 | . . . 4 | |
31 | 29, 30 | anim12i 321 | . . 3 |
32 | 31 | biantrurd 289 | . 2 |
33 | 3, 28, 32 | 3bitr4d 209 | 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 class class class wbr 3764 copab 3817 cxp 4343 |
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-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 |
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-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 |
This theorem is referenced by: ecopoveq 6201 oviec 6212 |
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