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Theorem ltsopi 8766
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 8750 . . . 4  |-  N.  =  ( om  \  { (/) } )
2 difss 3475 . . . . 5  |-  ( om 
\  { (/) } ) 
C_  om
3 omsson 4850 . . . . 5  |-  om  C_  On
42, 3sstri 3358 . . . 4  |-  ( om 
\  { (/) } ) 
C_  On
51, 4eqsstri 3379 . . 3  |-  N.  C_  On
6 epweon 4765 . . . 4  |-  _E  We  On
7 weso 4574 . . . 4  |-  (  _E  We  On  ->  _E  Or  On )
86, 7ax-mp 8 . . 3  |-  _E  Or  On
9 soss 4522 . . 3  |-  ( N.  C_  On  ->  (  _E  Or  On  ->  _E  Or  N. ) )
105, 8, 9mp2 9 . 2  |-  _E  Or  N.
11 df-lti 8753 . . . 4  |-  <N  =  (  _E  i^i  ( N.  X.  N. ) )
12 soeq1 4523 . . . 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 4943 . . 3  |-  (  _E  Or  N.  <->  (  _E  i^i  ( N.  X.  N. ) )  Or  N. )
1513, 14bitr4i 245 . 2  |-  (  <N  Or  N.  <->  _E  Or  N. )
1610, 15mpbir 202 1  |-  <N  Or  N.
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    \ cdif 3318    i^i cin 3320    C_ wss 3321   (/)c0 3629   {csn 3815    _E cep 4493    Or wor 4503    We wwe 4541   Oncon0 4582   omcom 4846    X. cxp 4877   N.cnpi 8720    <N clti 8723
This theorem is referenced by:  indpi  8785  nqereu  8807  ltsonq  8847  archnq  8858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-tr 4304  df-eprel 4495  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-ni 8750  df-lti 8753
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