Theorem axlttrn 7988

Description: Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7888 with ordering on the extended reals. New proofs should use lttr 7993 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 7888	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <RR B /\ B <RR C ) -> A <RR C ) )
2		ltxrlt 7985	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
3	2	3adant3 1012	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
4		ltxrlt 7985	. . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( B < C <-> B <RR C ) )
5	4	3adant1 1010	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B < C <-> B <RR C ) )
6	3, 5	anbi12d 470	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) <-> ( A <RR B /\ B <RR C ) ) )
7		ltxrlt 7985	. . 3 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
8	7	3adant2 1011	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> A <RR C ) )
9	1, 6, 8	3imtr4d 202	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   e. wcel 2141   class class class wbr 3989   RRcr 7773   <RR cltrr 7778   < clt 7954

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-lttrn 7888

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-pnf 7956  df-mnf 7957  df-ltxr 7959

This theorem is referenced by:  lttr  7993