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Mirrors > Home > ILE Home > Th. List > Mathboxes > if0ab | Unicode version |
Description: Expression of a
conditional class as a class abstraction when the False
alternative is the empty class: in that case, the conditional class is
the extension, in the True alternative, of the condition.
Remark: a consequence which could be formalized is the inclusion and therefore, using elpwg 3551, , from which fmelpw1o 13423 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.) |
Ref | Expression |
---|---|
if0ab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3398 | . . . . . 6 | |
2 | 1 | intnanr 916 | . . . . 5 |
3 | 2 | biorfi 736 | . . . 4 |
4 | 3 | bicomi 131 | . . 3 |
5 | 4 | abbii 2273 | . 2 |
6 | df-if 3506 | . 2 | |
7 | df-rab 2444 | . 2 | |
8 | 5, 6, 7 | 3eqtr4i 2188 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wo 698 wceq 1335 wcel 2128 cab 2143 crab 2439 c0 3394 cif 3505 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rab 2444 df-v 2714 df-dif 3104 df-nul 3395 df-if 3506 |
This theorem is referenced by: (None) |
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