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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 8579 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
reapirr | ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 8051 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | reapval 8550 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 #ℝ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 < 𝐴))) | |
3 | 2 | anidms 397 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 #ℝ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 < 𝐴))) |
4 | oridm 758 | . . 3 ⊢ ((𝐴 < 𝐴 ∨ 𝐴 < 𝐴) ↔ 𝐴 < 𝐴) | |
5 | 3, 4 | bitrdi 196 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 #ℝ 𝐴 ↔ 𝐴 < 𝐴)) |
6 | 1, 5 | mtbird 674 | 1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∨ wo 709 ∈ wcel 2159 class class class wbr 4017 ℝcr 7827 < clt 8009 #ℝ creap 8548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-pre-ltirr 7940 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-rab 2476 df-v 2753 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-br 4018 df-opab 4079 df-xp 4646 df-pnf 8011 df-mnf 8012 df-ltxr 8014 df-reap 8549 |
This theorem is referenced by: apirr 8579 |
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