Theorem reapti 8818

Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8861. (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 8315	. . . . 5 |- ( A e. RR -> -. A < A )
2	1	adantr 276	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> -. A < A )
3		oridm 765	. . . . . 6 |- ( ( A < A \/ A < A ) <-> A < A )
4		breq2 4097	. . . . . . 7 |- ( A = B -> ( A < A <-> A < B ) )
5		breq1 4096	. . . . . . 7 |- ( A = B -> ( A < A <-> B < A ) )
6	4, 5	orbi12d 801	. . . . . 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 673	. . . 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 8815	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )
11	10	notbid 673	. . 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 8309	. . . 4 |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )
14	13	3expia 1232	. . 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 716   = wceq 1398   e. wcel 2202   class class class wbr 4093   RRcr 8091   < clt 8273   #_ℝ creap 8813

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-pre-ltirr 8204  ax-pre-apti 8207

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-reap 8814

This theorem is referenced by:  rimul  8824  apreap  8826  apti  8861