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Theorem ltsopi 8512
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 8496 . . . 4  |-  N.  =  ( om  \  { (/) } )
2 difss 3303 . . . . 5  |-  ( om 
\  { (/) } ) 
C_  om
3 omsson 4660 . . . . 5  |-  om  C_  On
42, 3sstri 3188 . . . 4  |-  ( om 
\  { (/) } ) 
C_  On
51, 4eqsstri 3208 . . 3  |-  N.  C_  On
6 epweon 4575 . . . 4  |-  _E  We  On
7 weso 4384 . . . 4  |-  (  _E  We  On  ->  _E  Or  On )
86, 7ax-mp 8 . . 3  |-  _E  Or  On
9 soss 4332 . . 3  |-  ( N.  C_  On  ->  (  _E  Or  On  ->  _E  Or  N. ) )
105, 8, 9mp2 17 . 2  |-  _E  Or  N.
11 df-lti 8499 . . . 4  |-  <N  =  (  _E  i^i  ( N.  X.  N. ) )
12 soeq1 4333 . . . 4  |-  (  <N  =  (  _E  i^i  ( N.  X.  N. )
)  ->  (  <N  Or 
N. 
<->  (  _E  i^i  ( N.  X.  N. ) )  Or  N. ) )
1311, 12ax-mp 8 . . 3  |-  (  <N  Or  N.  <->  (  _E  i^i  ( N.  X.  N. )
)  Or  N. )
14 soinxp 4754 . . 3  |-  (  _E  Or  N.  <->  (  _E  i^i  ( N.  X.  N. ) )  Or  N. )
1513, 14bitr4i 243 . 2  |-  (  <N  Or  N.  <->  _E  Or  N. )
1610, 15mpbir 200 1  |-  <N  Or  N.
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640    _E cep 4303    Or wor 4313    We wwe 4351   Oncon0 4392   omcom 4656    X. cxp 4687   N.cnpi 8466    <N clti 8469
This theorem is referenced by:  indpi  8531  nqereu  8553  ltsonq  8593  archnq  8604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-ni 8496  df-lti 8499
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