Theorem reapti 7570

Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7613. (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 7095	. . . . 5 |- ( A e. RR -> -. A < A )
2	1	adantr 261	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> -. A < A )
3		oridm 674	. . . . . 6 |- ( ( A < A \/ A < A ) <-> A < A )
4		breq2 3768	. . . . . . 7 |- ( A = B -> ( A < A <-> A < B ) )
5		breq1 3767	. . . . . . 7 |- ( A = B -> ( A < A <-> B < A ) )
6	4, 5	orbi12d 707	. . . . . 6 |- ( A = B -> ( ( A < A \/ A < A ) <-> ( A < B \/ B < A ) ) )
7	3, 6	syl5bbr 183	. . . . 5 |- ( A = B -> ( A < A <-> ( A < B \/ B < A ) ) )
8	7	notbid 592	. . . 4 |- ( A = B -> ( -. A < A <-> -. ( A < B \/ B < A ) ) )
9	2, 8	syl5ibcom 144	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A = B -> -. ( A < B \/ B < A ) ) )
10		reapval 7567	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )
11	10	notbid 592	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. A #ℝ B <-> -. ( A < B \/ B < A ) ) )
12	9, 11	sylibrd 158	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A = B -> -. A #ℝ B ) )
13		axapti 7090	. . . 4 |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )
14	13	3expia 1106	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) -> A = B ) )
15	11, 14	sylbid 139	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( -. A #ℝ B -> A = B ) )
16	12, 15	impbid 120	1 |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #ℝ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   = wceq 1243   e. wcel 1393   class class class wbr 3764   RRcr 6888   < clt 7060   #_ℝ creap 7565

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996  ax-pre-apti 6999

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-ltxr 7065  df-reap 7566

This theorem is referenced by:  rimul  7576  apreap  7578  apti  7613