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Theorem reapval 7743
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.)
Assertion
Ref Expression
reapval ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))

Proof of Theorem reapval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 3798 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 < 𝑦𝐴 < 𝐵))
2 simpr 108 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
3 simpl 107 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
42, 3breq12d 3806 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 < 𝑥𝐵 < 𝐴))
51, 4orbi12d 740 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 < 𝑦𝑦 < 𝑥) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
6 df-reap 7742 . . 3 # = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦𝑦 < 𝑥))}
75, 6brab2ga 4441 . 2 (𝐴 # 𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 < 𝐵𝐵 < 𝐴)))
87baib 862 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 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  cr 7042   < clt 7215   # 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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