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Theorem reapval 7320
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7332 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
reapval ((A B ℝ) → (A # B ↔ (A < B B < A)))

Proof of Theorem reapval
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 3760 . . . 4 ((x = A y = B) → (x < yA < B))
2 simpr 103 . . . . 5 ((x = A y = B) → y = B)
3 simpl 102 . . . . 5 ((x = A y = B) → x = A)
42, 3breq12d 3768 . . . 4 ((x = A y = B) → (y < xB < A))
51, 4orbi12d 706 . . 3 ((x = A y = B) → ((x < y y < x) ↔ (A < B B < A)))
6 df-reap 7319 . . 3 # = {⟨x, y⟩ ∣ ((x y ℝ) (x < y y < x))}
75, 6brab2ga 4358 . 2 (A # B ↔ ((A B ℝ) (A < B B < A)))
87baib 827 1 ((A B ℝ) → (A # B ↔ (A < B B < A)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628   = wceq 1242   wcel 1390   class class class wbr 3755  cr 6670   < clt 6817   # creap 7318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-reap 7319
This theorem is referenced by:  reapirr  7321  recexre  7322  reapti  7323  reaplt  7332
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