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Theorem ltpiord 9317
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 9305 . . 3  |-  <N  =  (  _E  i^i  ( N.  X.  N. ) )
21breqi 4411 . 2  |-  ( A 
<N  B  <->  A (  _E  i^i  ( N.  X.  N. )
) B )
3 brinxp 4900 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  _E  B  <->  A (  _E  i^i  ( N.  X.  N. ) ) B ) )
4 epelg 4749 . . . 4  |-  ( B  e.  N.  ->  ( A  _E  B  <->  A  e.  B ) )
54adantl 468 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  _E  B  <->  A  e.  B ) )
63, 5bitr3d 259 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A (  _E 
i^i  ( N.  X.  N. ) ) B  <->  A  e.  B ) )
72, 6syl5bb 261 1  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    e. wcel 1889    i^i cin 3405   class class class wbr 4405    _E cep 4746    X. cxp 4835   N.cnpi 9274    <N clti 9277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-eprel 4748  df-xp 4843  df-lti 9305
This theorem is referenced by:  ltexpi  9332  ltapi  9333  ltmpi  9334  1lt2pi  9335  nlt1pi  9336  indpi  9337  nqereu  9359
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