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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 7730 | . 2 | |
2 | reapirr 8306 | . . . . . . . . . 10 #ℝ | |
3 | apreap 8316 | . . . . . . . . . . 11 # #ℝ | |
4 | 3 | anidms 394 | . . . . . . . . . 10 # #ℝ |
5 | 2, 4 | mtbird 647 | . . . . . . . . 9 # |
6 | reapirr 8306 | . . . . . . . . . 10 #ℝ | |
7 | apreap 8316 | . . . . . . . . . . 11 # #ℝ | |
8 | 7 | anidms 394 | . . . . . . . . . 10 # #ℝ |
9 | 6, 8 | mtbird 647 | . . . . . . . . 9 # |
10 | 5, 9 | anim12i 336 | . . . . . . . 8 # # |
11 | ioran 726 | . . . . . . . 8 # # # # | |
12 | 10, 11 | sylibr 133 | . . . . . . 7 # # |
13 | apreim 8332 | . . . . . . . 8 # # # | |
14 | 13 | anidms 394 | . . . . . . 7 # # # |
15 | 12, 14 | mtbird 647 | . . . . . 6 # |
16 | 15 | ad2antlr 480 | . . . . 5 # |
17 | id 19 | . . . . . . . 8 | |
18 | 17, 17 | breq12d 3912 | . . . . . . 7 # # |
19 | 18 | notbid 641 | . . . . . 6 # # |
20 | 19 | adantl 275 | . . . . 5 # # |
21 | 16, 20 | mpbird 166 | . . . 4 # |
22 | 21 | ex 114 | . . 3 # |
23 | 22 | rexlimdvva 2534 | . 2 # |
24 | 1, 23 | mpd 13 | 1 # |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 wceq 1316 wcel 1465 wrex 2394 class class class wbr 3899 (class class class)co 5742 cc 7586 cr 7587 ci 7590 caddc 7591 cmul 7593 #ℝ creap 8303 # cap 8310 |
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-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
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-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 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-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 |
This theorem is referenced by: mulap0r 8344 eirr 11397 |
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