Theorem reapval 8648

Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8660 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 4048	. . . 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 4056	. . . 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 8647	. . 3 |- #ℝ = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ ( x < y \/ y < x ) ) }
7	5, 6	brab2ga 4749	. 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 1372   e. wcel 2175   class class class wbr 4043   RRcr 7923   < clt 8106   #_ℝ creap 8646

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-reap 8647

This theorem is referenced by:  reapirr  8649  recexre  8650  reapti  8651  reaplt  8660