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Mirrors > Home > ILE Home > Th. List > reapti | Unicode version |
Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8352. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
reapti | #ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7809 | . . . . 5 | |
2 | 1 | adantr 274 | . . . 4 |
3 | oridm 731 | . . . . . 6 | |
4 | breq2 3903 | . . . . . . 7 | |
5 | breq1 3902 | . . . . . . 7 | |
6 | 4, 5 | orbi12d 767 | . . . . . 6 |
7 | 3, 6 | syl5bbr 193 | . . . . 5 |
8 | 7 | notbid 641 | . . . 4 |
9 | 2, 8 | syl5ibcom 154 | . . 3 |
10 | reapval 8306 | . . . 4 #ℝ | |
11 | 10 | notbid 641 | . . 3 #ℝ |
12 | 9, 11 | sylibrd 168 | . 2 #ℝ |
13 | axapti 7803 | . . . 4 | |
14 | 13 | 3expia 1168 | . . 3 |
15 | 11, 14 | sylbid 149 | . 2 #ℝ |
16 | 12, 15 | impbid 128 | 1 #ℝ |
Colors of variables: wff set class |
Syntax hints: wn 3 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-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-13 1476 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 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 ax-pre-apti 7703 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 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-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-reap 8305 |
This theorem is referenced by: rimul 8315 apreap 8317 apti 8352 |
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