Theorem axapti 7859

Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7759 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)

Assertion

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

Proof of Theorem axapti

Step	Hyp	Ref	Expression
1		ltxrlt 7854	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
2		ltxrlt 7854	. . . . . 6 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> B <RR A ) )
3	2	ancoms 266	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( B < A <-> B <RR A ) )
4	1, 3	orbi12d 783	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ B < A ) <-> ( A <RR B \/ B <RR A ) ) )
5	4	notbid 657	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) <-> -. ( A <RR B \/ B <RR A ) ) )
6		ax-pre-apti 7759	. . . 4 |- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) -> A = B )
7	6	3expia 1184	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A <RR B \/ B <RR A ) -> A = B ) )
8	5, 7	sylbid 149	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) -> A = B ) )
9	8	3impia 1179	1 |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 963   = wceq 1332   e. wcel 1481   class class class wbr 3937   RRcr 7643   <RR cltrr 7648   < clt 7824

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-pre-apti 7759

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-pnf 7826  df-mnf 7827  df-ltxr 7829

This theorem is referenced by:  lttri3  7868  reapti  8365