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Theorem axlttrn 8771
Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttrn 8689 with ordering on the extended reals.) (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
StepHypRef Expression
1 ax-pre-lttrn 8689 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <RR  B  /\  B  <RR  C )  ->  A  <RR  C ) )
2 ltxrlt 8769 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
323adant3 980 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
4 ltxrlt 8769 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  <  C  <->  B 
<RR  C ) )
543adant1 978 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  C  <->  B  <RR  C ) )
63, 5anbi12d 694 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <  C )  <-> 
( A  <RR  B  /\  B  <RR  C ) ) )
7 ltxrlt 8769 . . 3  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  A 
<RR  C ) )
873adant2 979 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  A  <RR  C ) )
91, 6, 83imtr4d 261 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    e. wcel 1621   class class class wbr 3917   RRcr 8613    <RR cltrr 8618    < clt 8744
This theorem is referenced by:  lttr  8775  ltso  8779  lelttr  8788  ltletr  8789  lttri  8821  lttrd  8853  mulgt1  9463  recgt1i  9501  recreclt  9503  sup2  9558  nnge1  9620  recnz  9933  gtndiv  9935  xrlttr  10320  expnbnd  11072  sin01gt0  12306  cos01gt0  12307  chtub  20217  lvsovso  24721
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4035  ax-nul 4043  ax-pow 4079  ax-pr 4105  ax-un 4400  ax-resscn 8671  ax-pre-lttrn 8689
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2511  df-rex 2512  df-rab 2514  df-v 2727  df-sbc 2920  df-csb 3007  df-dif 3078  df-un 3080  df-in 3082  df-ss 3086  df-nul 3360  df-if 3468  df-pw 3529  df-sn 3547  df-pr 3548  df-op 3550  df-uni 3725  df-br 3918  df-opab 3972  df-mpt 3973  df-id 4199  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-fun 4599  df-fn 4600  df-f 4601  df-f1 4602  df-fo 4603  df-f1o 4604  df-fv 4605  df-er 6543  df-en 6747  df-dom 6748  df-sdom 6749  df-pnf 8746  df-mnf 8747  df-ltxr 8749
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