Theorem reapti 7679

Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7722. (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 7188	. . . . 5 |- ( A e. RR -> -. A < A )
2	1	adantr 270	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> -. A < A )
3		oridm 706	. . . . . 6 |- ( ( A < A \/ A < A ) <-> A < A )
4		breq2 3789	. . . . . . 7 |- ( A = B -> ( A < A <-> A < B ) )
5		breq1 3788	. . . . . . 7 |- ( A = B -> ( A < A <-> B < A ) )
6	4, 5	orbi12d 739	. . . . . 6 |- ( A = B -> ( ( A < A \/ A < A ) <-> ( A < B \/ B < A ) ) )
7	3, 6	syl5bbr 192	. . . . 5 |- ( A = B -> ( A < A <-> ( A < B \/ B < A ) ) )
8	7	notbid 624	. . . 4 |- ( A = B -> ( -. A < A <-> -. ( A < B \/ B < A ) ) )
9	2, 8	syl5ibcom 153	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A = B -> -. ( A < B \/ B < A ) ) )
10		reapval 7676	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )
11	10	notbid 624	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. A #ℝ B <-> -. ( A < B \/ B < A ) ) )
12	9, 11	sylibrd 167	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A = B -> -. A #ℝ B ) )
13		axapti 7183	. . . 4 |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )
14	13	3expia 1140	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) -> A = B ) )
15	11, 14	sylbid 148	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( -. A #ℝ B -> A = B ) )
16	12, 15	impbid 127	1 |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #ℝ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   class class class wbr 3785   RRcr 6980   < clt 7153   #_ℝ creap 7674

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-pre-ltirr 7088  ax-pre-apti 7091

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-pnf 7155  df-mnf 7156  df-ltxr 7158  df-reap 7675

This theorem is referenced by:  rimul  7685  apreap  7687  apti  7722