Theorem axapti 8090

Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7987 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 8085	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
2		ltxrlt 8085	. . . . . 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 794	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ B < A ) <-> ( A <RR B \/ B <RR A ) ) )
5	4	notbid 668	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) <-> -. ( A <RR B \/ B <RR A ) ) )
6		ax-pre-apti 7987	. . . 4 |- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) -> A = B )
7	6	3expia 1207	. . 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 1202	1 |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4029   RRcr 7871   <RR cltrr 7876   < clt 8054

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-pre-apti 7987

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-pnf 8056  df-mnf 8057  df-ltxr 8059

This theorem is referenced by:  lttri3  8099  reapti  8598