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Theorem ltsopi 9312
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 9296 . . . 4  |-  N.  =  ( om  \  { (/) } )
2 difss 3598 . . . . 5  |-  ( om 
\  { (/) } ) 
C_  om
3 omsson 6710 . . . . 5  |-  om  C_  On
42, 3sstri 3479 . . . 4  |-  ( om 
\  { (/) } ) 
C_  On
51, 4eqsstri 3500 . . 3  |-  N.  C_  On
6 epweon 6624 . . . 4  |-  _E  We  On
7 weso 4845 . . . 4  |-  (  _E  We  On  ->  _E  Or  On )
86, 7ax-mp 5 . . 3  |-  _E  Or  On
9 soss 4793 . . 3  |-  ( N.  C_  On  ->  (  _E  Or  On  ->  _E  Or  N. ) )
105, 8, 9mp2 9 . 2  |-  _E  Or  N.
11 df-lti 9299 . . . 4  |-  <N  =  (  _E  i^i  ( N.  X.  N. ) )
12 soeq1 4794 . . . 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 4919 . . 3  |-  (  _E  Or  N.  <->  (  _E  i^i  ( N.  X.  N. ) )  Or  N. )
1513, 14bitr4i 255 . 2  |-  (  <N  Or  N.  <->  _E  Or  N. )
1610, 15mpbir 212 1  |-  <N  Or  N.
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    \ cdif 3439    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002    _E cep 4763    Or wor 4774    We wwe 4812    X. cxp 4852   Oncon0 5442   omcom 6706   N.cnpi 9268    <N clti 9271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-om 6707  df-ni 9296  df-lti 9299
This theorem is referenced by:  indpi  9331  nqereu  9353  ltsonq  9393  archnq  9404
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