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Mirrors > Home > ILE Home > Th. List > reapti | GIF version |
Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8384. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
reapti | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 #ℝ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7841 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | 1 | adantr 274 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ 𝐴 < 𝐴) |
3 | oridm 746 | . . . . . 6 ⊢ ((𝐴 < 𝐴 ∨ 𝐴 < 𝐴) ↔ 𝐴 < 𝐴) | |
4 | breq2 3933 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵)) | |
5 | breq1 3932 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐵 < 𝐴)) | |
6 | 4, 5 | orbi12d 782 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐴 < 𝐴 ∨ 𝐴 < 𝐴) ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
7 | 3, 6 | syl5bbr 193 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
8 | 7 | notbid 656 | . . . 4 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 < 𝐴 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
9 | 2, 8 | syl5ibcom 154 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
10 | reapval 8338 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 #ℝ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | |
11 | 10 | notbid 656 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 #ℝ 𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
12 | 9, 11 | sylibrd 168 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → ¬ 𝐴 #ℝ 𝐵)) |
13 | axapti 7835 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) → 𝐴 = 𝐵) | |
14 | 13 | 3expia 1183 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴) → 𝐴 = 𝐵)) |
15 | 11, 14 | sylbid 149 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 #ℝ 𝐵 → 𝐴 = 𝐵)) |
16 | 12, 15 | impbid 128 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 #ℝ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ℝcr 7619 < clt 7800 #ℝ creap 8336 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 ax-pre-apti 7735 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-reap 8337 |
This theorem is referenced by: rimul 8347 apreap 8349 apti 8384 |
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