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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 8480 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
reapirr | ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7954 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | reapval 8451 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 #ℝ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 < 𝐴))) | |
3 | 2 | anidms 395 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 #ℝ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 < 𝐴))) |
4 | oridm 747 | . . 3 ⊢ ((𝐴 < 𝐴 ∨ 𝐴 < 𝐴) ↔ 𝐴 < 𝐴) | |
5 | 3, 4 | bitrdi 195 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 #ℝ 𝐴 ↔ 𝐴 < 𝐴)) |
6 | 1, 5 | mtbird 663 | 1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 698 ∈ wcel 2128 class class class wbr 3965 ℝcr 7731 < clt 7912 #ℝ creap 8449 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-pre-ltirr 7844 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4592 df-pnf 7914 df-mnf 7915 df-ltxr 7917 df-reap 8450 |
This theorem is referenced by: apirr 8480 |
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