Theorem axapti 8228

Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 8125 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 8223	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
2		ltxrlt 8223	. . . . . 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 798	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ B < A ) <-> ( A <RR B \/ B <RR A ) ) )
5	4	notbid 671	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A < B \/ B < A ) <-> -. ( A <RR B \/ B <RR A ) ) )
6		ax-pre-apti 8125	. . . 4 |- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) -> A = B )
7	6	3expia 1229	. . 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 1224	1 |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083   RRcr 8009   <RR cltrr 8014   < clt 8192

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-pre-apti 8125

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8194  df-mnf 8195  df-ltxr 8197

This theorem is referenced by:  lttri3  8237  reapti  8737