Theorem reapval 8603

Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8615 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)

Assertion

Ref	Expression
reapval	|- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )

Proof of Theorem reapval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq12 4038	. . . 4 |- ( ( x = A /\ y = B ) -> ( x < y <-> A < B ) )
2		simpr 110	. . . . 5 |- ( ( x = A /\ y = B ) -> y = B )
3		simpl 109	. . . . 5 |- ( ( x = A /\ y = B ) -> x = A )
4	2, 3	breq12d 4046	. . . 4 |- ( ( x = A /\ y = B ) -> ( y < x <-> B < A ) )
5	1, 4	orbi12d 794	. . 3 |- ( ( x = A /\ y = B ) -> ( ( x < y \/ y < x ) <-> ( A < B \/ B < A ) ) )
6		df-reap 8602	. . 3 |- #ℝ = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ ( x < y \/ y < x ) ) }
7	5, 6	brab2ga 4738	. 2 |- ( A #ℝ B <-> ( ( A e. RR /\ B e. RR ) /\ ( A < B \/ B < A ) ) )
8	7	baib 920	1 |- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2167   class class class wbr 4033   RRcr 7878   < clt 8061   #_ℝ creap 8601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-reap 8602

This theorem is referenced by:  reapirr  8604  recexre  8605  reapti  8606  reaplt  8615