Theorem axapti 8386

Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 8284 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 8381	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
2		ltxrlt 8381	. . . . . 6 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> B <RR A ) )
3	2	ancoms 268	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( B < A <-> B <RR A ) )
4	1, 3	orbi12d 805	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ B < A ) <-> ( A <RR B \/ B <RR A ) ) )
5	4	notbid 677	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) <-> -. ( A <RR B \/ B <RR A ) ) )
6		ax-pre-apti 8284	. . . 4 |- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) -> A = B )
7	6	3expia 1236	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A <RR B \/ B <RR A ) -> A = B ) )
8	5, 7	sylbid 150	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) -> A = B ) )
9	8	3impia 1231	1 |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   RRcr 8168   <RR cltrr 8173   < clt 8350

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-apti 8284

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-pnf 8352  df-mnf 8353  df-ltxr 8355

This theorem is referenced by:  lttri3  8395  reapti  8897