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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 3910 | . . . . . . 7 |
5 | 4 | notbid 641 | . . . . . 6 |
6 | 2, 5 | raleqbidv 2615 | . . . . 5 |
7 | 3 | breqd 3910 | . . . . . . 7 |
8 | 3 | breqd 3910 | . . . . . . . 8 |
9 | 2, 8 | rexeqbidv 2616 | . . . . . . 7 |
10 | 7, 9 | imbi12d 233 | . . . . . 6 |
11 | 1, 10 | raleqbidv 2615 | . . . . 5 |
12 | 6, 11 | anbi12d 464 | . . . 4 |
13 | 1, 12 | rabeqbidv 2655 | . . 3 |
14 | 13 | unieqd 3717 | . 2 |
15 | df-sup 6839 | . 2 | |
16 | df-sup 6839 | . 2 | |
17 | 14, 15, 16 | 3eqtr4g 2175 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1316 wral 2393 wrex 2394 crab 2397 cuni 3706 class class class wbr 3899 csup 6837 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-uni 3707 df-br 3900 df-sup 6839 |
This theorem is referenced by: infeq123d 6871 |
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