Theorem axltwlin 8247

Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 8145 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 8145	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( A <RR C \/ C <RR B ) ) )
2		ltxrlt 8245	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
3	2	3adant3 1043	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
4		ltxrlt 8245	. . . 4 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
5	4	3adant2 1042	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
6		ltxrlt 8245	. . . . 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 1041	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C < B <-> C <RR B ) )
9	5, 8	orbi12d 800	. 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 715   /\ w3a 1004   e. wcel 2202   class class class wbr 4088   RRcr 8031   <RR cltrr 8036   < clt 8214

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-pre-ltwlin 8145

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-pnf 8216  df-mnf 8217  df-ltxr 8219

This theorem is referenced by:  ltso  8257  letr  8262  lelttr  8268  ltletr  8269  gt0add  8753  reapcotr  8778  sup3exmid  9137  xrltso  10031  rebtwn2zlemstep  10513  expnbnd  10926  leabs  11652  ltabs  11665  abslt  11666  absle  11667  maxabslemlub  11785  suplociccreex  15367  ivthinclemloc  15384  ivthdichlem  15394  cnplimclemle  15411  reeff1o  15516  efltlemlt  15517  sin0pilem2  15525  coseq0negpitopi  15579  cos02pilt1  15594