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Theorem reapval 7611
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7623 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 3794 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 < 𝑦𝐴 < 𝐵))
2 simpr 107 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
3 simpl 106 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
42, 3breq12d 3802 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 < 𝑥𝐵 < 𝐴))
51, 4orbi12d 715 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 < 𝑦𝑦 < 𝑥) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
6 df-reap 7610 . . 3 # = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦𝑦 < 𝑥))}
75, 6brab2ga 4440 . 2 (𝐴 # 𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 < 𝐵𝐵 < 𝐴)))
87baib 837 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 637   = wceq 1257  wcel 1407   class class class wbr 3789  cr 6916   < clt 7089   # creap 7609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-xp 4376  df-reap 7610
This theorem is referenced by:  reapirr  7612  recexre  7613  reapti  7614  reaplt  7623
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