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Theorem ltsopi 8528
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 8512 . . . 4  |-  N.  =  ( om  \  { (/) } )
2 difss 3316 . . . . 5  |-  ( om 
\  { (/) } ) 
C_  om
3 omsson 4676 . . . . 5  |-  om  C_  On
42, 3sstri 3201 . . . 4  |-  ( om 
\  { (/) } ) 
C_  On
51, 4eqsstri 3221 . . 3  |-  N.  C_  On
6 epweon 4591 . . . 4  |-  _E  We  On
7 weso 4400 . . . 4  |-  (  _E  We  On  ->  _E  Or  On )
86, 7ax-mp 8 . . 3  |-  _E  Or  On
9 soss 4348 . . 3  |-  ( N.  C_  On  ->  (  _E  Or  On  ->  _E  Or  N. ) )
105, 8, 9mp2 17 . 2  |-  _E  Or  N.
11 df-lti 8515 . . . 4  |-  <N  =  (  _E  i^i  ( N.  X.  N. ) )
12 soeq1 4349 . . . 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 4770 . . 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 1632    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653    _E cep 4319    Or wor 4329    We wwe 4367   Oncon0 4408   omcom 4672    X. cxp 4703   N.cnpi 8482    <N clti 8485
This theorem is referenced by:  indpi  8547  nqereu  8569  ltsonq  8609  archnq  8620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-ni 8512  df-lti 8515
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