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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 1893 | . . . . . 6 | |
2 | an12 551 | . . . . . . 7 | |
3 | 2 | exbii 1592 | . . . . . 6 |
4 | elxp 4615 | . . . . . . . 8 | |
5 | excom 1651 | . . . . . . . . 9 | |
6 | an12 551 | . . . . . . . . . . . . 13 | |
7 | velsn 3587 | . . . . . . . . . . . . . 14 | |
8 | 7 | anbi1i 454 | . . . . . . . . . . . . 13 |
9 | 6, 8 | bitri 183 | . . . . . . . . . . . 12 |
10 | 9 | exbii 1592 | . . . . . . . . . . 11 |
11 | vex 2724 | . . . . . . . . . . . 12 | |
12 | opeq1 3752 | . . . . . . . . . . . . . 14 | |
13 | 12 | eqeq2d 2176 | . . . . . . . . . . . . 13 |
14 | 13 | anbi1d 461 | . . . . . . . . . . . 12 |
15 | 11, 14 | ceqsexv 2760 | . . . . . . . . . . 11 |
16 | 10, 15 | bitri 183 | . . . . . . . . . 10 |
17 | 16 | exbii 1592 | . . . . . . . . 9 |
18 | 5, 17 | bitri 183 | . . . . . . . 8 |
19 | nfv 1515 | . . . . . . . . . 10 | |
20 | nfsab1 2154 | . . . . . . . . . 10 | |
21 | 19, 20 | nfan 1552 | . . . . . . . . 9 |
22 | nfv 1515 | . . . . . . . . 9 | |
23 | opeq2 3753 | . . . . . . . . . . 11 | |
24 | 23 | eqeq2d 2176 | . . . . . . . . . 10 |
25 | df-clab 2151 | . . . . . . . . . . 11 | |
26 | sbequ12 1758 | . . . . . . . . . . . 12 | |
27 | 26 | equcoms 1695 | . . . . . . . . . . 11 |
28 | 25, 27 | bitr4id 198 | . . . . . . . . . 10 |
29 | 24, 28 | anbi12d 465 | . . . . . . . . 9 |
30 | 21, 22, 29 | cbvex 1743 | . . . . . . . 8 |
31 | 4, 18, 30 | 3bitri 205 | . . . . . . 7 |
32 | 31 | anbi2i 453 | . . . . . 6 |
33 | 1, 3, 32 | 3bitr4ri 212 | . . . . 5 |
34 | 33 | exbii 1592 | . . . 4 |
35 | eliun 3864 | . . . . 5 | |
36 | df-rex 2448 | . . . . 5 | |
37 | 35, 36 | bitri 183 | . . . 4 |
38 | elopab 4230 | . . . 4 | |
39 | 34, 37, 38 | 3bitr4i 211 | . . 3 |
40 | 39 | eqriv 2161 | . 2 |
41 | opabex3d.1 | . . 3 | |
42 | snexg 4157 | . . . . . 6 | |
43 | 11, 42 | ax-mp 5 | . . . . 5 |
44 | opabex3d.2 | . . . . 5 | |
45 | xpexg 4712 | . . . . 5 | |
46 | 43, 44, 45 | sylancr 411 | . . . 4 |
47 | 46 | ralrimiva 2537 | . . 3 |
48 | iunexg 6079 | . . 3 | |
49 | 41, 47, 48 | syl2anc 409 | . 2 |
50 | 40, 49 | eqeltrrid 2252 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wsb 1749 wcel 2135 cab 2150 wral 2442 wrex 2443 cvv 2721 csn 3570 cop 3573 ciun 3860 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-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
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-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 |
This theorem is referenced by: acfun 7154 ccfunen 7196 ovshftex 10747 |
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