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Mirrors > Home > ILE Home > Th. List > apti | Unicode version |
Description: Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.) |
Ref | Expression |
---|---|
apti | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 7765 | . . 3 | |
2 | 1 | adantr 274 | . 2 |
3 | cnre 7765 | . . . . . . 7 | |
4 | 3 | adantl 275 | . . . . . 6 |
5 | 4 | ad2antrr 479 | . . . . 5 |
6 | simpr 109 | . . . . . . . . . 10 | |
7 | 6 | ad3antrrr 483 | . . . . . . . . 9 |
8 | simplr 519 | . . . . . . . . 9 | |
9 | cru 8367 | . . . . . . . . 9 | |
10 | 7, 8, 9 | syl2anc 408 | . . . . . . . 8 |
11 | simpllr 523 | . . . . . . . . 9 | |
12 | simpr 109 | . . . . . . . . 9 | |
13 | 11, 12 | eqeq12d 2154 | . . . . . . . 8 |
14 | apreim 8368 | . . . . . . . . . . . 12 # # # | |
15 | 14 | notbid 656 | . . . . . . . . . . 11 # # # |
16 | ioran 741 | . . . . . . . . . . 11 # # # # | |
17 | 15, 16 | syl6bb 195 | . . . . . . . . . 10 # # # |
18 | 7, 8, 17 | syl2anc 408 | . . . . . . . . 9 # # # |
19 | 11, 12 | breq12d 3942 | . . . . . . . . . 10 # # |
20 | 19 | notbid 656 | . . . . . . . . 9 # # |
21 | 7 | simpld 111 | . . . . . . . . . . 11 |
22 | 8 | simpld 111 | . . . . . . . . . . 11 |
23 | reapti 8344 | . . . . . . . . . . . 12 #ℝ | |
24 | apreap 8352 | . . . . . . . . . . . . 13 # #ℝ | |
25 | 24 | notbid 656 | . . . . . . . . . . . 12 # #ℝ |
26 | 23, 25 | bitr4d 190 | . . . . . . . . . . 11 # |
27 | 21, 22, 26 | syl2anc 408 | . . . . . . . . . 10 # |
28 | 7 | simprd 113 | . . . . . . . . . . 11 |
29 | 8 | simprd 113 | . . . . . . . . . . 11 |
30 | reapti 8344 | . . . . . . . . . . . 12 #ℝ | |
31 | apreap 8352 | . . . . . . . . . . . . 13 # #ℝ | |
32 | 31 | notbid 656 | . . . . . . . . . . . 12 # #ℝ |
33 | 30, 32 | bitr4d 190 | . . . . . . . . . . 11 # |
34 | 28, 29, 33 | syl2anc 408 | . . . . . . . . . 10 # |
35 | 27, 34 | anbi12d 464 | . . . . . . . . 9 # # |
36 | 18, 20, 35 | 3bitr4d 219 | . . . . . . . 8 # |
37 | 10, 13, 36 | 3bitr4d 219 | . . . . . . 7 # |
38 | 37 | ex 114 | . . . . . 6 # |
39 | 38 | rexlimdvva 2557 | . . . . 5 # |
40 | 5, 39 | mpd 13 | . . . 4 # |
41 | 40 | ex 114 | . . 3 # |
42 | 41 | rexlimdvva 2557 | . 2 # |
43 | 2, 42 | mpd 13 | 1 # |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 wrex 2417 class class class wbr 3929 (class class class)co 5774 cc 7621 cr 7622 ci 7625 caddc 7626 cmul 7628 #ℝ creap 8339 # cap 8346 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-ltxr 7808 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 |
This theorem is referenced by: apne 8388 apcon4bid 8389 cnstab 8410 qapne 9434 expeq0 10327 nn0opthd 10471 recvguniq 10770 climuni 11065 dedekindeu 12773 dedekindicclemicc 12782 ivthinc 12793 limcimo 12806 cnplimclemle 12809 coseq0q4123 12918 refeq 13226 triap 13227 |
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