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Theorem reapval 7163
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7175 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 3732 . . . 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 3740 . . . 4 ((x = A y = B) → (y < xB < A))
51, 4orbi12d 691 . . 3 ((x = A y = B) → ((x < y y < x) ↔ (A < B B < A)))
6 df-reap 7162 . . 3 # = {⟨x, y⟩ ∣ ((x y ℝ) (x < y y < x))}
75, 6brab2ga 4330 . 2 (A # B ↔ ((A B ℝ) (A < B B < A)))
87baib 812 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 613   = wceq 1223   wcel 1366   class class class wbr 3727  cr 6519   < clt 6660   # creap 7161
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-xp 4266  df-reap 7162
This theorem is referenced by:  reapirr  7164  recexre  7165  reapti  7166  reaplt  7175
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