Theorem reapval 8597

Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8609 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 4035	. . . 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 4043	. . . 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 8596	. . 3 |- #ℝ = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ ( x < y \/ y < x ) ) }
7	5, 6	brab2ga 4735	. 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 2164   class class class wbr 4030   RRcr 7873   < clt 8056   #_ℝ creap 8595

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-reap 8596

This theorem is referenced by:  reapirr  8598  recexre  8599  reapti  8600  reaplt  8609