Theorem axltwlin 7966

Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7866 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 7866	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( A <RR C \/ C <RR B ) ) )
2		ltxrlt 7964	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
3	2	3adant3 1007	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
4		ltxrlt 7964	. . . 4 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
5	4	3adant2 1006	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
6		ltxrlt 7964	. . . . 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 1005	. . 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 968   e. wcel 2136   class class class wbr 3982   RRcr 7752   <RR cltrr 7757   < clt 7933

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-ltwlin 7866

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-pnf 7935  df-mnf 7936  df-ltxr 7938

This theorem is referenced by:  ltso  7976  letr  7981  lelttr  7987  ltletr  7988  gt0add  8471  reapcotr  8496  sup3exmid  8852  xrltso  9732  rebtwn2zlemstep  10188  expnbnd  10578  leabs  11016  ltabs  11029  abslt  11030  absle  11031  maxabslemlub  11149  suplociccreex  13242  ivthinclemloc  13259  cnplimclemle  13277  reeff1o  13334  efltlemlt  13335  sin0pilem2  13343  coseq0negpitopi  13397  cos02pilt1  13412