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Mirrors > Home > ILE Home > Th. List > reapirr | Unicode version |
Description: Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8360 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
reapirr | #ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7834 | . 2 | |
2 | reapval 8331 | . . . 4 #ℝ | |
3 | 2 | anidms 394 | . . 3 #ℝ |
4 | oridm 746 | . . 3 | |
5 | 3, 4 | syl6bb 195 | . 2 #ℝ |
6 | 1, 5 | mtbird 662 | 1 #ℝ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wo 697 wcel 1480 class class class wbr 3924 cr 7612 clt 7793 #ℝ creap 8329 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-reap 8330 |
This theorem is referenced by: apirr 8360 |
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