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Theorem ltpiord 9061
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 9049 . . 3  |-  <N  =  (  _E  i^i  ( N.  X.  N. ) )
21breqi 4303 . 2  |-  ( A 
<N  B  <->  A (  _E  i^i  ( N.  X.  N. )
) B )
3 brinxp 4906 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  _E  B  <->  A (  _E  i^i  ( N.  X.  N. ) ) B ) )
4 epelg 4638 . . . 4  |-  ( B  e.  N.  ->  ( A  _E  B  <->  A  e.  B ) )
54adantl 466 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  _E  B  <->  A  e.  B ) )
63, 5bitr3d 255 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A (  _E 
i^i  ( N.  X.  N. ) ) B  <->  A  e.  B ) )
72, 6syl5bb 257 1  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756    i^i cin 3332   class class class wbr 4297    _E cep 4635    X. cxp 4843   N.cnpi 9016    <N clti 9019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-eprel 4637  df-xp 4851  df-lti 9049
This theorem is referenced by:  ltexpi  9076  ltapi  9077  ltmpi  9078  1lt2pi  9079  nlt1pi  9080  indpi  9081  nqereu  9103
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