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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 7115 | . . 3 | |
2 | 1 | adantr 270 | . 2 |
3 | cnre 7115 | . . . . . . 7 | |
4 | 3 | adantl 271 | . . . . . 6 |
5 | 4 | ad2antrr 471 | . . . . 5 |
6 | simpr 108 | . . . . . . . . . 10 | |
7 | 6 | ad3antrrr 475 | . . . . . . . . 9 |
8 | simplr 496 | . . . . . . . . 9 | |
9 | cru 7702 | . . . . . . . . 9 | |
10 | 7, 8, 9 | syl2anc 403 | . . . . . . . 8 |
11 | simpllr 500 | . . . . . . . . 9 | |
12 | simpr 108 | . . . . . . . . 9 | |
13 | 11, 12 | eqeq12d 2095 | . . . . . . . 8 |
14 | apreim 7703 | . . . . . . . . . . . 12 # # # | |
15 | 14 | notbid 624 | . . . . . . . . . . 11 # # # |
16 | ioran 701 | . . . . . . . . . . 11 # # # # | |
17 | 15, 16 | syl6bb 194 | . . . . . . . . . 10 # # # |
18 | 7, 8, 17 | syl2anc 403 | . . . . . . . . 9 # # # |
19 | 11, 12 | breq12d 3798 | . . . . . . . . . 10 # # |
20 | 19 | notbid 624 | . . . . . . . . 9 # # |
21 | 7 | simpld 110 | . . . . . . . . . . 11 |
22 | 8 | simpld 110 | . . . . . . . . . . 11 |
23 | reapti 7679 | . . . . . . . . . . . 12 #ℝ | |
24 | apreap 7687 | . . . . . . . . . . . . 13 # #ℝ | |
25 | 24 | notbid 624 | . . . . . . . . . . . 12 # #ℝ |
26 | 23, 25 | bitr4d 189 | . . . . . . . . . . 11 # |
27 | 21, 22, 26 | syl2anc 403 | . . . . . . . . . 10 # |
28 | 7 | simprd 112 | . . . . . . . . . . 11 |
29 | 8 | simprd 112 | . . . . . . . . . . 11 |
30 | reapti 7679 | . . . . . . . . . . . 12 #ℝ | |
31 | apreap 7687 | . . . . . . . . . . . . 13 # #ℝ | |
32 | 31 | notbid 624 | . . . . . . . . . . . 12 # #ℝ |
33 | 30, 32 | bitr4d 189 | . . . . . . . . . . 11 # |
34 | 28, 29, 33 | syl2anc 403 | . . . . . . . . . 10 # |
35 | 27, 34 | anbi12d 456 | . . . . . . . . 9 # # |
36 | 18, 20, 35 | 3bitr4d 218 | . . . . . . . 8 # |
37 | 10, 13, 36 | 3bitr4d 218 | . . . . . . 7 # |
38 | 37 | ex 113 | . . . . . 6 # |
39 | 38 | rexlimdvva 2484 | . . . . 5 # |
40 | 5, 39 | mpd 13 | . . . 4 # |
41 | 40 | ex 113 | . . 3 # |
42 | 41 | rexlimdvva 2484 | . 2 # |
43 | 2, 42 | mpd 13 | 1 # |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 wceq 1284 wcel 1433 wrex 2349 class class class wbr 3785 (class class class)co 5532 cc 6979 cr 6980 ci 6983 caddc 6984 cmul 6986 #ℝ creap 7674 # cap 7681 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 |
This theorem is referenced by: apne 7723 qapne 8724 expeq0 9507 nn0opthd 9649 recvguniq 9881 abs00 9950 climuni 10132 |
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