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Theorem lttri4d 9619
Description: Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Hypotheses
Ref Expression
ltd.1  |-  ( ph  ->  A  e.  RR )
ltd.2  |-  ( ph  ->  B  e.  RR )
Assertion
Ref Expression
lttri4d  |-  ( ph  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )

Proof of Theorem lttri4d
StepHypRef Expression
1 ltd.1 . 2  |-  ( ph  ->  A  e.  RR )
2 ltd.2 . 2  |-  ( ph  ->  B  e.  RR )
3 lttri4 9563 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
41, 2, 3syl2anc 661 1  |-  ( ph  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 964    = wceq 1370    e. wcel 1758   class class class wbr 4393   RRcr 9385    < clt 9522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-resscn 9443  ax-pre-lttri 9460  ax-pre-lttrn 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-ltxr 9527
This theorem is referenced by:  icccvx  20647  ivthicc  21067  dvivth  21608  coseq00topi  22090  cvxcl  22504  scvxcvx  22505
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