Theorem reapti 8900

Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8943. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)

Assertion

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

Proof of Theorem reapti

Step	Hyp	Ref	Expression
1		ltnr 8395	. . . . 5 |- ( A e. RR -> -. A < A )
2	1	adantr 276	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> -. A < A )
3		oridm 769	. . . . . 6 |- ( ( A < A \/ A < A ) <-> A < A )
4		breq2 4132	. . . . . . 7 |- ( A = B -> ( A < A <-> A < B ) )
5		breq1 4131	. . . . . . 7 |- ( A = B -> ( A < A <-> B < A ) )
6	4, 5	orbi12d 805	. . . . . 6 |- ( A = B -> ( ( A < A \/ A < A ) <-> ( A < B \/ B < A ) ) )
7	3, 6	bitr3id 194	. . . . 5 |- ( A = B -> ( A < A <-> ( A < B \/ B < A ) ) )
8	7	notbid 677	. . . 4 |- ( A = B -> ( -. A < A <-> -. ( A < B \/ B < A ) ) )
9	2, 8	syl5ibcom 155	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A = B -> -. ( A < B \/ B < A ) ) )
10		reapval 8897	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )
11	10	notbid 677	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. A #ℝ B <-> -. ( A < B \/ B < A ) ) )
12	9, 11	sylibrd 169	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A = B -> -. A #ℝ B ) )
13		axapti 8389	. . . 4 |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )
14	13	3expia 1236	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) -> A = B ) )
15	11, 14	sylbid 150	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( -. A #ℝ B -> A = B ) )
16	12, 15	impbid 129	1 |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #ℝ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   class class class wbr 4128   RRcr 8171   < clt 8353   #_ℝ creap 8895

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-pre-ltirr 8284  ax-pre-apti 8287

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-xp 4778  df-pnf 8355  df-mnf 8356  df-ltxr 8358  df-reap 8896

This theorem is referenced by:  rimul  8906  apreap  8908  apti  8943