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Mirrors > Home > MPE Home > Th. List > supeq123d | Structured version Visualization version 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 4664 | . . . . . . 7 |
5 | 4 | notbid 308 | . . . . . 6 |
6 | 2, 5 | raleqbidv 3152 | . . . . 5 |
7 | 3 | breqd 4664 | . . . . . . 7 |
8 | 3 | breqd 4664 | . . . . . . . 8 |
9 | 2, 8 | rexeqbidv 3153 | . . . . . . 7 |
10 | 7, 9 | imbi12d 334 | . . . . . 6 |
11 | 1, 10 | raleqbidv 3152 | . . . . 5 |
12 | 6, 11 | anbi12d 747 | . . . 4 |
13 | 1, 12 | rabeqbidv 3195 | . . 3 |
14 | 13 | unieqd 4446 | . 2 |
15 | df-sup 8348 | . 2 | |
16 | df-sup 8348 | . 2 | |
17 | 14, 15, 16 | 3eqtr4g 2681 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wral 2912 wrex 2913 crab 2916 cuni 4436 class class class wbr 4653 csup 8346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-uni 4437 df-br 4654 df-sup 8348 |
This theorem is referenced by: infeq123d 8387 wlimeq12 31765 aomclem8 37631 |
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