Theorem reapval 8679

Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8691 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 4059	. . . 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 4067	. . . 4 |- ( ( x = A /\ y = B ) -> ( y < x <-> B < A ) )
5	1, 4	orbi12d 795	. . 3 |- ( ( x = A /\ y = B ) -> ( ( x < y \/ y < x ) <-> ( A < B \/ B < A ) ) )
6		df-reap 8678	. . 3 |- #ℝ = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ ( x < y \/ y < x ) ) }
7	5, 6	brab2ga 4763	. 2 |- ( A #ℝ B <-> ( ( A e. RR /\ B e. RR ) /\ ( A < B \/ B < A ) ) )
8	7	baib 921	1 |- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2177   class class class wbr 4054   RRcr 7954   < clt 8137   #_ℝ creap 8677

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4055  df-opab 4117  df-xp 4694  df-reap 8678

This theorem is referenced by:  reapirr  8680  recexre  8681  reapti  8682  reaplt  8691