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Mirrors > Home > ILE Home > Th. List > reapval | Unicode version |
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8318 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
Ref | Expression |
---|---|
reapval | #ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq12 3904 | . . . 4 | |
2 | simpr 109 | . . . . 5 | |
3 | simpl 108 | . . . . 5 | |
4 | 2, 3 | breq12d 3912 | . . . 4 |
5 | 1, 4 | orbi12d 767 | . . 3 |
6 | df-reap 8305 | . . 3 #ℝ | |
7 | 5, 6 | brab2ga 4584 | . 2 #ℝ |
8 | 7 | baib 889 | 1 #ℝ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 wceq 1316 wcel 1465 class class class wbr 3899 cr 7587 clt 7768 #ℝ creap 8304 |
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-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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-reap 8305 |
This theorem is referenced by: reapirr 8307 recexre 8308 reapti 8309 reaplt 8318 |
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