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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq 4005 |
. . . . . . 7
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2 | 1 | notbid 667 |
. . . . . 6
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3 | 2 | ralbidv 2477 |
. . . . 5
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4 | breq 4005 |
. . . . . . 7
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5 | breq 4005 |
. . . . . . . 8
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6 | 5 | rexbidv 2478 |
. . . . . . 7
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7 | 4, 6 | imbi12d 234 |
. . . . . 6
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8 | 7 | ralbidv 2477 |
. . . . 5
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9 | 3, 8 | anbi12d 473 |
. . . 4
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10 | 9 | rabbidv 2726 |
. . 3
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11 | 10 | unieqd 3820 |
. 2
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12 | df-sup 6982 |
. 2
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13 | df-sup 6982 |
. 2
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14 | 11, 12, 13 | 3eqtr4g 2235 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-uni 3810 df-br 4004 df-sup 6982 |
This theorem is referenced by: infeq3 7013 |
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