Theorem axlttrn 8211

Description: Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 8109 with ordering on the extended reals. New proofs should use lttr 8216 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Assertion

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

Proof of Theorem axlttrn

Step	Hyp	Ref	Expression
1		ax-pre-lttrn 8109	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <RR B /\ B <RR C ) -> A <RR C ) )
2		ltxrlt 8208	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
3	2	3adant3 1041	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
4		ltxrlt 8208	. . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( B < C <-> B <RR C ) )
5	4	3adant1 1039	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B < C <-> B <RR C ) )
6	3, 5	anbi12d 473	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) <-> ( A <RR B /\ B <RR C ) ) )
7		ltxrlt 8208	. . 3 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
8	7	3adant2 1040	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
9	1, 6, 8	3imtr4d 203	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   e. wcel 2200   class class class wbr 4082   RRcr 7994   <RR cltrr 7999   < clt 8177

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-pre-lttrn 8109

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-pnf 8179  df-mnf 8180  df-ltxr 8182

This theorem is referenced by:  lttr  8216