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Theorem reapval 8361
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8373 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 3941 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 < 𝑦𝐴 < 𝐵))
2 simpr 109 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
3 simpl 108 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
42, 3breq12d 3949 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 < 𝑥𝐵 < 𝐴))
51, 4orbi12d 783 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 < 𝑦𝑦 < 𝑥) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
6 df-reap 8360 . . 3 # = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦𝑦 < 𝑥))}
75, 6brab2ga 4621 . 2 (𝐴 # 𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 < 𝐵𝐵 < 𝐴)))
87baib 905 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 698   = wceq 1332  wcel 1481   class class class wbr 3936  cr 7642   < clt 7823   # creap 8359
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-reap 8360
This theorem is referenced by:  reapirr  8362  recexre  8363  reapti  8364  reaplt  8373
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