Theorem axltwlin 7856

Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7757 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 7757	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( A <RR C \/ C <RR B ) ) )
2		ltxrlt 7854	. . 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 7854	. . . 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 7854	. . . . 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 1481   class class class wbr 3937   RRcr 7643   <RR cltrr 7648   < clt 7824

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-pre-ltwlin 7757

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-pnf 7826  df-mnf 7827  df-ltxr 7829

This theorem is referenced by:  ltso  7866  letr  7871  lelttr  7876  ltletr  7877  gt0add  8359  reapcotr  8384  sup3exmid  8739  xrltso  9612  rebtwn2zlemstep  10061  expnbnd  10446  leabs  10878  ltabs  10891  abslt  10892  absle  10893  maxabslemlub  11011  suplociccreex  12810  ivthinclemloc  12827  cnplimclemle  12845  reeff1o  12902  efltlemlt  12903  sin0pilem2  12911  coseq0negpitopi  12965  cos02pilt1  12980