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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 8524 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
reapirr | ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7996 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | reapval 8495 | . . . 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 752 | . . 3 ⊢ ((𝐴 < 𝐴 ∨ 𝐴 < 𝐴) ↔ 𝐴 < 𝐴) | |
5 | 3, 4 | bitrdi 195 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 #ℝ 𝐴 ↔ 𝐴 < 𝐴)) |
6 | 1, 5 | mtbird 668 | 1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 703 ∈ wcel 2141 class class class wbr 3989 ℝcr 7773 < clt 7954 #ℝ creap 8493 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltirr 7886 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-ltxr 7959 df-reap 8494 |
This theorem is referenced by: apirr 8524 |
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