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Mirrors > Home > ILE Home > Th. List > opabex3 | Unicode version |
Description: Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
opabex3.1 | |
opabex3.2 |
Ref | Expression |
---|---|
opabex3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.42v 1899 | . . . . . 6 | |
2 | an12 556 | . . . . . . 7 | |
3 | 2 | exbii 1598 | . . . . . 6 |
4 | elxp 4628 | . . . . . . . 8 | |
5 | excom 1657 | . . . . . . . . 9 | |
6 | an12 556 | . . . . . . . . . . . . 13 | |
7 | velsn 3600 | . . . . . . . . . . . . . 14 | |
8 | 7 | anbi1i 455 | . . . . . . . . . . . . 13 |
9 | 6, 8 | bitri 183 | . . . . . . . . . . . 12 |
10 | 9 | exbii 1598 | . . . . . . . . . . 11 |
11 | vex 2733 | . . . . . . . . . . . 12 | |
12 | opeq1 3765 | . . . . . . . . . . . . . 14 | |
13 | 12 | eqeq2d 2182 | . . . . . . . . . . . . 13 |
14 | 13 | anbi1d 462 | . . . . . . . . . . . 12 |
15 | 11, 14 | ceqsexv 2769 | . . . . . . . . . . 11 |
16 | 10, 15 | bitri 183 | . . . . . . . . . 10 |
17 | 16 | exbii 1598 | . . . . . . . . 9 |
18 | 5, 17 | bitri 183 | . . . . . . . 8 |
19 | nfv 1521 | . . . . . . . . . 10 | |
20 | nfsab1 2160 | . . . . . . . . . 10 | |
21 | 19, 20 | nfan 1558 | . . . . . . . . 9 |
22 | nfv 1521 | . . . . . . . . 9 | |
23 | opeq2 3766 | . . . . . . . . . . 11 | |
24 | 23 | eqeq2d 2182 | . . . . . . . . . 10 |
25 | df-clab 2157 | . . . . . . . . . . 11 | |
26 | sbequ12 1764 | . . . . . . . . . . . 12 | |
27 | 26 | equcoms 1701 | . . . . . . . . . . 11 |
28 | 25, 27 | bitr4id 198 | . . . . . . . . . 10 |
29 | 24, 28 | anbi12d 470 | . . . . . . . . 9 |
30 | 21, 22, 29 | cbvex 1749 | . . . . . . . 8 |
31 | 4, 18, 30 | 3bitri 205 | . . . . . . 7 |
32 | 31 | anbi2i 454 | . . . . . 6 |
33 | 1, 3, 32 | 3bitr4ri 212 | . . . . 5 |
34 | 33 | exbii 1598 | . . . 4 |
35 | eliun 3877 | . . . . 5 | |
36 | df-rex 2454 | . . . . 5 | |
37 | 35, 36 | bitri 183 | . . . 4 |
38 | elopab 4243 | . . . 4 | |
39 | 34, 37, 38 | 3bitr4i 211 | . . 3 |
40 | 39 | eqriv 2167 | . 2 |
41 | opabex3.1 | . . 3 | |
42 | snexg 4170 | . . . . . 6 | |
43 | 11, 42 | ax-mp 5 | . . . . 5 |
44 | opabex3.2 | . . . . 5 | |
45 | xpexg 4725 | . . . . 5 | |
46 | 43, 44, 45 | sylancr 412 | . . . 4 |
47 | 46 | rgen 2523 | . . 3 |
48 | iunexg 6098 | . . 3 | |
49 | 41, 47, 48 | mp2an 424 | . 2 |
50 | 40, 49 | eqeltrri 2244 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wex 1485 wsb 1755 wcel 2141 cab 2156 wral 2448 wrex 2449 cvv 2730 csn 3583 cop 3586 ciun 3873 copab 4049 cxp 4609 |
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-coll 4104 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-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 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-iun 3875 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-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 |
This theorem is referenced by: (None) |
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