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Mirrors > Home > ILE Home > Th. List > supeq123d | Unicode version |
Description: Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
Ref | Expression |
---|---|
supeq123d.a | |
supeq123d.b | |
supeq123d.c |
Ref | Expression |
---|---|
supeq123d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supeq123d.b | . . . 4 | |
2 | supeq123d.a | . . . . . 6 | |
3 | supeq123d.c | . . . . . . . 8 | |
4 | 3 | breqd 3998 | . . . . . . 7 |
5 | 4 | notbid 662 | . . . . . 6 |
6 | 2, 5 | raleqbidv 2677 | . . . . 5 |
7 | 3 | breqd 3998 | . . . . . . 7 |
8 | 3 | breqd 3998 | . . . . . . . 8 |
9 | 2, 8 | rexeqbidv 2678 | . . . . . . 7 |
10 | 7, 9 | imbi12d 233 | . . . . . 6 |
11 | 1, 10 | raleqbidv 2677 | . . . . 5 |
12 | 6, 11 | anbi12d 470 | . . . 4 |
13 | 1, 12 | rabeqbidv 2725 | . . 3 |
14 | 13 | unieqd 3805 | . 2 |
15 | df-sup 6959 | . 2 | |
16 | df-sup 6959 | . 2 | |
17 | 14, 15, 16 | 3eqtr4g 2228 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1348 wral 2448 wrex 2449 crab 2452 cuni 3794 class class class wbr 3987 csup 6957 |
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-in1 609 ax-in2 610 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-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-uni 3795 df-br 3988 df-sup 6959 |
This theorem is referenced by: infeq123d 6991 |
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