Theorem axltwlin 7946

Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7846 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)

Assertion

Ref	Expression
axltwlin	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( A < C \/ C < B ) ) )

Proof of Theorem axltwlin

Step	Hyp	Ref	Expression
1		ax-pre-ltwlin 7846	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( A <RR C \/ C <RR B ) ) )
2		ltxrlt 7944	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
3	2	3adant3 1002	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
4		ltxrlt 7944	. . . 4 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
5	4	3adant2 1001	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
6		ltxrlt 7944	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C < B <-> C <RR B ) )
7	6	ancoms 266	. . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( C < B <-> C <RR B ) )
8	7	3adant1 1000	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C < B <-> C <RR B ) )
9	5, 8	orbi12d 783	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < C \/ C < B ) <-> ( A <RR C \/ C <RR B ) ) )
10	1, 3, 9	3imtr4d 202	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( A < C \/ C < B ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 698   /\ w3a 963   e. wcel 2128   class class class wbr 3966   RRcr 7732   <RR cltrr 7737   < clt 7913

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-cnex 7824  ax-resscn 7825  ax-pre-ltwlin 7846

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-xp 4593  df-pnf 7915  df-mnf 7916  df-ltxr 7918

This theorem is referenced by:  ltso  7956  letr  7961  lelttr  7966  ltletr  7967  gt0add  8449  reapcotr  8474  sup3exmid  8829  xrltso  9704  rebtwn2zlemstep  10156  expnbnd  10545  leabs  10978  ltabs  10991  abslt  10992  absle  10993  maxabslemlub  11111  suplociccreex  13044  ivthinclemloc  13061  cnplimclemle  13079  reeff1o  13136  efltlemlt  13137  sin0pilem2  13145  coseq0negpitopi  13199  cos02pilt1  13214