Theorem axapti 8156

Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 8053 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 8151	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
2		ltxrlt 8151	. . . . . 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 795	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ B < A ) <-> ( A <RR B \/ B <RR A ) ) )
5	4	notbid 669	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) <-> -. ( A <RR B \/ B <RR A ) ) )
6		ax-pre-apti 8053	. . . 4 |- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) -> A = B )
7	6	3expia 1208	. . 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 1203	1 |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2177   class class class wbr 4048   RRcr 7937   <RR cltrr 7942   < clt 8120

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-pre-apti 8053

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-xp 4686  df-pnf 8122  df-mnf 8123  df-ltxr 8125

This theorem is referenced by:  lttri3  8165  reapti  8665