Theorem axlttrn 8290

Description: Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 8189 with ordering on the extended reals. New proofs should use lttr 8295 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 8189	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <RR B /\ B <RR C ) -> A <RR C ) )
2		ltxrlt 8287	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
3	2	3adant3 1044	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
4		ltxrlt 8287	. . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( B < C <-> B <RR C ) )
5	4	3adant1 1042	. . 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 8287	. . 3 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
8	7	3adant2 1043	. 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 1005   e. wcel 2202   class class class wbr 4093   RRcr 8074   <RR cltrr 8079   < clt 8256

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-pre-lttrn 8189

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-pnf 8258  df-mnf 8259  df-ltxr 8261

This theorem is referenced by:  lttr  8295