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Mirrors > Home > ILE Home > Th. List > opabex3d | Unicode version |
Description: Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.) |
Ref | Expression |
---|---|
opabex3d.1 | |
opabex3d.2 |
Ref | Expression |
---|---|
opabex3d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.42v 1878 | . . . . . 6 | |
2 | an12 550 | . . . . . . 7 | |
3 | 2 | exbii 1584 | . . . . . 6 |
4 | elxp 4551 | . . . . . . . 8 | |
5 | excom 1642 | . . . . . . . . 9 | |
6 | an12 550 | . . . . . . . . . . . . 13 | |
7 | velsn 3539 | . . . . . . . . . . . . . 14 | |
8 | 7 | anbi1i 453 | . . . . . . . . . . . . 13 |
9 | 6, 8 | bitri 183 | . . . . . . . . . . . 12 |
10 | 9 | exbii 1584 | . . . . . . . . . . 11 |
11 | vex 2684 | . . . . . . . . . . . 12 | |
12 | opeq1 3700 | . . . . . . . . . . . . . 14 | |
13 | 12 | eqeq2d 2149 | . . . . . . . . . . . . 13 |
14 | 13 | anbi1d 460 | . . . . . . . . . . . 12 |
15 | 11, 14 | ceqsexv 2720 | . . . . . . . . . . 11 |
16 | 10, 15 | bitri 183 | . . . . . . . . . 10 |
17 | 16 | exbii 1584 | . . . . . . . . 9 |
18 | 5, 17 | bitri 183 | . . . . . . . 8 |
19 | nfv 1508 | . . . . . . . . . 10 | |
20 | nfsab1 2127 | . . . . . . . . . 10 | |
21 | 19, 20 | nfan 1544 | . . . . . . . . 9 |
22 | nfv 1508 | . . . . . . . . 9 | |
23 | opeq2 3701 | . . . . . . . . . . 11 | |
24 | 23 | eqeq2d 2149 | . . . . . . . . . 10 |
25 | sbequ12 1744 | . . . . . . . . . . . 12 | |
26 | 25 | equcoms 1684 | . . . . . . . . . . 11 |
27 | df-clab 2124 | . . . . . . . . . . 11 | |
28 | 26, 27 | syl6rbbr 198 | . . . . . . . . . 10 |
29 | 24, 28 | anbi12d 464 | . . . . . . . . 9 |
30 | 21, 22, 29 | cbvex 1729 | . . . . . . . 8 |
31 | 4, 18, 30 | 3bitri 205 | . . . . . . 7 |
32 | 31 | anbi2i 452 | . . . . . 6 |
33 | 1, 3, 32 | 3bitr4ri 212 | . . . . 5 |
34 | 33 | exbii 1584 | . . . 4 |
35 | eliun 3812 | . . . . 5 | |
36 | df-rex 2420 | . . . . 5 | |
37 | 35, 36 | bitri 183 | . . . 4 |
38 | elopab 4175 | . . . 4 | |
39 | 34, 37, 38 | 3bitr4i 211 | . . 3 |
40 | 39 | eqriv 2134 | . 2 |
41 | opabex3d.1 | . . 3 | |
42 | snexg 4103 | . . . . . 6 | |
43 | 11, 42 | ax-mp 5 | . . . . 5 |
44 | opabex3d.2 | . . . . 5 | |
45 | xpexg 4648 | . . . . 5 | |
46 | 43, 44, 45 | sylancr 410 | . . . 4 |
47 | 46 | ralrimiva 2503 | . . 3 |
48 | iunexg 6010 | . . 3 | |
49 | 41, 47, 48 | syl2anc 408 | . 2 |
50 | 40, 49 | eqeltrrid 2225 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wsb 1735 cab 2123 wral 2414 wrex 2415 cvv 2681 csn 3522 cop 3525 ciun 3808 copab 3983 cxp 4532 |
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-coll 4038 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-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 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-iun 3810 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-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 |
This theorem is referenced by: acfun 7056 ccfunen 7072 ovshftex 10584 |
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