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Mirrors > Home > ILE Home > Th. List > reapirr | GIF 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 7705 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
reapirr | ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7188 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | reapval 7676 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 #ℝ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 < 𝐴))) | |
3 | 2 | anidms 389 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 #ℝ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 < 𝐴))) |
4 | oridm 706 | . . 3 ⊢ ((𝐴 < 𝐴 ∨ 𝐴 < 𝐴) ↔ 𝐴 < 𝐴) | |
5 | 3, 4 | syl6bb 194 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 #ℝ 𝐴 ↔ 𝐴 < 𝐴)) |
6 | 1, 5 | mtbird 630 | 1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 ∨ wo 661 ∈ wcel 1433 class class class wbr 3785 ℝcr 6980 < clt 7153 #ℝ creap 7674 |
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-pre-ltirr 7088 |
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-rab 2357 df-v 2603 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-xp 4369 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-reap 7675 |
This theorem is referenced by: apirr 7705 |
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