Theorem axltwlin 8043

Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7942 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 7942	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( A <RR C \/ C <RR B ) ) )
2		ltxrlt 8041	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
3	2	3adant3 1019	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
4		ltxrlt 8041	. . . 4 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
5	4	3adant2 1018	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
6		ltxrlt 8041	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C < B <-> C <RR B ) )
7	6	ancoms 268	. . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( C < B <-> C <RR B ) )
8	7	3adant1 1017	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C < B <-> C <RR B ) )
9	5, 8	orbi12d 794	. 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 203	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( A < C \/ C < B ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 709   /\ w3a 980   e. wcel 2160   class class class wbr 4018   RRcr 7828   <RR cltrr 7833   < clt 8010

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-pre-ltwlin 7942

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4647  df-pnf 8012  df-mnf 8013  df-ltxr 8015

This theorem is referenced by:  ltso  8053  letr  8058  lelttr  8064  ltletr  8065  gt0add  8548  reapcotr  8573  sup3exmid  8932  xrltso  9814  rebtwn2zlemstep  10271  expnbnd  10662  leabs  11101  ltabs  11114  abslt  11115  absle  11116  maxabslemlub  11234  suplociccreex  14499  ivthinclemloc  14516  cnplimclemle  14534  reeff1o  14591  efltlemlt  14592  sin0pilem2  14600  coseq0negpitopi  14654  cos02pilt1  14669