Theorem reapval 8746

Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8758 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 4091	. . . 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 4099	. . . 4 |- ( ( x = A /\ y = B ) -> ( y < x <-> B < A ) )
5	1, 4	orbi12d 798	. . 3 |- ( ( x = A /\ y = B ) -> ( ( x < y \/ y < x ) <-> ( A < B \/ B < A ) ) )
6		df-reap 8745	. . 3 |- #ℝ = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ ( x < y \/ y < x ) ) }
7	5, 6	brab2ga 4799	. 2 |- ( A #ℝ B <-> ( ( A e. RR /\ B e. RR ) /\ ( A < B \/ B < A ) ) )
8	7	baib 924	1 |- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   class class class wbr 4086   RRcr 8021   < clt 8204   #_ℝ creap 8744

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-reap 8745

This theorem is referenced by:  reapirr  8747  recexre  8748  reapti  8749  reaplt  8758