Theorem reapti 8344

Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8387. (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 7844	. . . . 5 |- ( A e. RR -> -. A < A )
2	1	adantr 274	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> -. A < A )
3		oridm 746	. . . . . 6 |- ( ( A < A \/ A < A ) <-> A < A )
4		breq2 3933	. . . . . . 7 |- ( A = B -> ( A < A <-> A < B ) )
5		breq1 3932	. . . . . . 7 |- ( A = B -> ( A < A <-> B < A ) )
6	4, 5	orbi12d 782	. . . . . 6 |- ( A = B -> ( ( A < A \/ A < A ) <-> ( A < B \/ B < A ) ) )
7	3, 6	syl5bbr 193	. . . . 5 |- ( A = B -> ( A < A <-> ( A < B \/ B < A ) ) )
8	7	notbid 656	. . . 4 |- ( A = B -> ( -. A < A <-> -. ( A < B \/ B < A ) ) )
9	2, 8	syl5ibcom 154	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A = B -> -. ( A < B \/ B < A ) ) )
10		reapval 8341	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )
11	10	notbid 656	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. A #ℝ B <-> -. ( A < B \/ B < A ) ) )
12	9, 11	sylibrd 168	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A = B -> -. A #ℝ B ) )
13		axapti 7838	. . . 4 |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )
14	13	3expia 1183	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) -> A = B ) )
15	11, 14	sylbid 149	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( -. A #ℝ B -> A = B ) )
16	12, 15	impbid 128	1 |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #ℝ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   class class class wbr 3929   RRcr 7622   < clt 7803   #_ℝ creap 8339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7714  ax-resscn 7715  ax-pre-ltirr 7735  ax-pre-apti 7738

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-pnf 7805  df-mnf 7806  df-ltxr 7808  df-reap 8340

This theorem is referenced by:  rimul  8350  apreap  8352  apti  8387