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Mirrors > Home > ILE Home > Th. List > ssoprab2b | Unicode version |
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4031. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
ssoprab2b |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfoprab1 5574 | . . . 4 | |
2 | nfoprab1 5574 | . . . 4 | |
3 | 1, 2 | nfss 2992 | . . 3 |
4 | nfoprab2 5575 | . . . . 5 | |
5 | nfoprab2 5575 | . . . . 5 | |
6 | 4, 5 | nfss 2992 | . . . 4 |
7 | nfoprab3 5576 | . . . . . 6 | |
8 | nfoprab3 5576 | . . . . . 6 | |
9 | 7, 8 | nfss 2992 | . . . . 5 |
10 | ssel 2993 | . . . . . 6 | |
11 | oprabid 5557 | . . . . . 6 | |
12 | oprabid 5557 | . . . . . 6 | |
13 | 10, 11, 12 | 3imtr3g 202 | . . . . 5 |
14 | 9, 13 | alrimi 1455 | . . . 4 |
15 | 6, 14 | alrimi 1455 | . . 3 |
16 | 3, 15 | alrimi 1455 | . 2 |
17 | ssoprab2 5581 | . 2 | |
18 | 16, 17 | impbii 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 103 wal 1282 wcel 1433 wss 2973 cop 3401 coprab 5533 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-oprab 5536 |
This theorem is referenced by: eqoprab2b 5583 |
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