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Mirrors > Home > ILE Home > Th. List > reapval | GIF version |
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7755 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
Ref | Expression |
---|---|
reapval | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 #ℝ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq12 3798 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 < 𝑦 ↔ 𝐴 < 𝐵)) | |
2 | simpr 108 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) | |
3 | simpl 107 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴) | |
4 | 2, 3 | breq12d 3806 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 < 𝑥 ↔ 𝐵 < 𝐴)) |
5 | 1, 4 | orbi12d 740 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
6 | df-reap 7742 | . . 3 ⊢ #ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))} | |
7 | 5, 6 | brab2ga 4441 | . 2 ⊢ (𝐴 #ℝ 𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
8 | 7 | baib 862 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 #ℝ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 662 = wceq 1285 ∈ wcel 1434 class class class wbr 3793 ℝcr 7042 < clt 7215 #ℝ creap 7741 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ral 2354 df-rex 2355 df-v 2604 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-br 3794 df-opab 3848 df-xp 4377 df-reap 7742 |
This theorem is referenced by: reapirr 7744 recexre 7745 reapti 7746 reaplt 7755 |
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