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Theorem ltpiord 8527
Description: Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
ltpiord  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B ) )

Proof of Theorem ltpiord
StepHypRef Expression
1 df-lti 8515 . . 3  |-  <N  =  (  _E  i^i  ( N.  X.  N. ) )
21breqi 4045 . 2  |-  ( A 
<N  B  <->  A (  _E  i^i  ( N.  X.  N. )
) B )
3 brinxp 4768 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  _E  B  <->  A (  _E  i^i  ( N.  X.  N. ) ) B ) )
4 epelg 4322 . . . 4  |-  ( B  e.  N.  ->  ( A  _E  B  <->  A  e.  B ) )
54adantl 452 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  _E  B  <->  A  e.  B ) )
63, 5bitr3d 246 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A (  _E 
i^i  ( N.  X.  N. ) ) B  <->  A  e.  B ) )
72, 6syl5bb 248 1  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696    i^i cin 3164   class class class wbr 4039    _E cep 4319    X. cxp 4703   N.cnpi 8482    <N clti 8485
This theorem is referenced by:  ltexpi  8542  ltapi  8543  ltmpi  8544  1lt2pi  8545  nlt1pi  8546  indpi  8547  nqereu  8569
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-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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-eprel 4321  df-xp 4711  df-lti 8515
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