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Theorem ltsopi 9055
Description: Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
Assertion
Ref Expression
ltsopi  |-  <N  Or  N.

Proof of Theorem ltsopi
StepHypRef Expression
1 df-ni 9039 . . . 4  |-  N.  =  ( om  \  { (/) } )
2 difss 3481 . . . . 5  |-  ( om 
\  { (/) } ) 
C_  om
3 omsson 6478 . . . . 5  |-  om  C_  On
42, 3sstri 3363 . . . 4  |-  ( om 
\  { (/) } ) 
C_  On
51, 4eqsstri 3384 . . 3  |-  N.  C_  On
6 epweon 6393 . . . 4  |-  _E  We  On
7 weso 4709 . . . 4  |-  (  _E  We  On  ->  _E  Or  On )
86, 7ax-mp 5 . . 3  |-  _E  Or  On
9 soss 4657 . . 3  |-  ( N.  C_  On  ->  (  _E  Or  On  ->  _E  Or  N. ) )
105, 8, 9mp2 9 . 2  |-  _E  Or  N.
11 df-lti 9042 . . . 4  |-  <N  =  (  _E  i^i  ( N.  X.  N. ) )
12 soeq1 4658 . . . 4  |-  (  <N  =  (  _E  i^i  ( N.  X.  N. )
)  ->  (  <N  Or 
N. 
<->  (  _E  i^i  ( N.  X.  N. ) )  Or  N. ) )
1311, 12ax-mp 5 . . 3  |-  (  <N  Or  N.  <->  (  _E  i^i  ( N.  X.  N. )
)  Or  N. )
14 soinxp 4901 . . 3  |-  (  _E  Or  N.  <->  (  _E  i^i  ( N.  X.  N. ) )  Or  N. )
1513, 14bitr4i 252 . 2  |-  (  <N  Or  N.  <->  _E  Or  N. )
1610, 15mpbir 209 1  |-  <N  Or  N.
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    \ cdif 3323    i^i cin 3325    C_ wss 3326   (/)c0 3635   {csn 3875    _E cep 4628    Or wor 4638    We wwe 4676   Oncon0 4717    X. cxp 4836   omcom 6474   N.cnpi 9009    <N clti 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-tr 4384  df-eprel 4630  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-om 6475  df-ni 9039  df-lti 9042
This theorem is referenced by:  indpi  9074  nqereu  9096  ltsonq  9136  archnq  9147
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