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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 7596 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
reapirr | ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7095 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | reapval 7567 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 #ℝ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 < 𝐴))) | |
3 | 2 | anidms 377 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 #ℝ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 < 𝐴))) |
4 | oridm 674 | . . 3 ⊢ ((𝐴 < 𝐴 ∨ 𝐴 < 𝐴) ↔ 𝐴 < 𝐴) | |
5 | 3, 4 | syl6bb 185 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 #ℝ 𝐴 ↔ 𝐴 < 𝐴)) |
6 | 1, 5 | mtbird 598 | 1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∨ wo 629 ∈ wcel 1393 class class class wbr 3764 ℝcr 6888 < clt 7060 #ℝ creap 7565 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-ltxr 7065 df-reap 7566 |
This theorem is referenced by: apirr 7596 |
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