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Mirrors > Home > ILE Home > Th. List > supeq3 | Unicode version |
Description: Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.) |
Ref | Expression |
---|---|
supeq3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq 3931 | . . . . . . 7 | |
2 | 1 | notbid 656 | . . . . . 6 |
3 | 2 | ralbidv 2437 | . . . . 5 |
4 | breq 3931 | . . . . . . 7 | |
5 | breq 3931 | . . . . . . . 8 | |
6 | 5 | rexbidv 2438 | . . . . . . 7 |
7 | 4, 6 | imbi12d 233 | . . . . . 6 |
8 | 7 | ralbidv 2437 | . . . . 5 |
9 | 3, 8 | anbi12d 464 | . . . 4 |
10 | 9 | rabbidv 2675 | . . 3 |
11 | 10 | unieqd 3747 | . 2 |
12 | df-sup 6871 | . 2 | |
13 | df-sup 6871 | . 2 | |
14 | 11, 12, 13 | 3eqtr4g 2197 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1331 wral 2416 wrex 2417 crab 2420 cuni 3736 class class class wbr 3929 csup 6869 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-uni 3737 df-br 3930 df-sup 6871 |
This theorem is referenced by: infeq3 6902 |
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