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Mirrors > Home > ILE Home > Th. List > apirr | Unicode version |
Description: Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
Ref | Expression |
---|---|
apirr | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 7886 | . 2 | |
2 | reapirr 8466 | . . . . . . . . . 10 #ℝ | |
3 | apreap 8476 | . . . . . . . . . . 11 # #ℝ | |
4 | 3 | anidms 395 | . . . . . . . . . 10 # #ℝ |
5 | 2, 4 | mtbird 663 | . . . . . . . . 9 # |
6 | reapirr 8466 | . . . . . . . . . 10 #ℝ | |
7 | apreap 8476 | . . . . . . . . . . 11 # #ℝ | |
8 | 7 | anidms 395 | . . . . . . . . . 10 # #ℝ |
9 | 6, 8 | mtbird 663 | . . . . . . . . 9 # |
10 | 5, 9 | anim12i 336 | . . . . . . . 8 # # |
11 | ioran 742 | . . . . . . . 8 # # # # | |
12 | 10, 11 | sylibr 133 | . . . . . . 7 # # |
13 | apreim 8492 | . . . . . . . 8 # # # | |
14 | 13 | anidms 395 | . . . . . . 7 # # # |
15 | 12, 14 | mtbird 663 | . . . . . 6 # |
16 | 15 | ad2antlr 481 | . . . . 5 # |
17 | id 19 | . . . . . . . 8 | |
18 | 17, 17 | breq12d 3989 | . . . . . . 7 # # |
19 | 18 | notbid 657 | . . . . . 6 # # |
20 | 19 | adantl 275 | . . . . 5 # # |
21 | 16, 20 | mpbird 166 | . . . 4 # |
22 | 21 | ex 114 | . . 3 # |
23 | 22 | rexlimdvva 2589 | . 2 # |
24 | 1, 23 | mpd 13 | 1 # |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 wrex 2443 class class class wbr 3976 (class class class)co 5836 cc 7742 cr 7743 ci 7746 caddc 7747 cmul 7749 #ℝ creap 8463 # cap 8470 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-ltxr 7929 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 |
This theorem is referenced by: mulap0r 8504 eirr 11705 dcapnconst 13773 |
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