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Mirrors > Home > ILE Home > Th. List > dfopab2 | Unicode version |
Description: A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfopab2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsbc1v 2964 | . . . . 5 | |
2 | 1 | 19.41 1673 | . . . 4 |
3 | sbcopeq1a 6147 | . . . . . . . 8 | |
4 | 3 | pm5.32i 450 | . . . . . . 7 |
5 | 4 | exbii 1592 | . . . . . 6 |
6 | nfcv 2306 | . . . . . . . 8 | |
7 | nfsbc1v 2964 | . . . . . . . 8 | |
8 | 6, 7 | nfsbc 2966 | . . . . . . 7 |
9 | 8 | 19.41 1673 | . . . . . 6 |
10 | 5, 9 | bitr3i 185 | . . . . 5 |
11 | 10 | exbii 1592 | . . . 4 |
12 | elvv 4660 | . . . . 5 | |
13 | 12 | anbi1i 454 | . . . 4 |
14 | 2, 11, 13 | 3bitr4i 211 | . . 3 |
15 | 14 | abbii 2280 | . 2 |
16 | df-opab 4038 | . 2 | |
17 | df-rab 2451 | . 2 | |
18 | 15, 16, 17 | 3eqtr4i 2195 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1342 wex 1479 wcel 2135 cab 2150 crab 2446 cvv 2721 wsbc 2946 cop 3573 copab 4036 cxp 4596 cfv 5182 c1st 6098 c2nd 6099 |
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-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-rab 2451 df-v 2723 df-sbc 2947 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-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-iota 5147 df-fun 5184 df-fv 5190 df-1st 6100 df-2nd 6101 |
This theorem is referenced by: (None) |
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