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Theorem ltpiord 7134
Description: Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
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 7122 . . 3  |-  <N  =  (  _E  i^i  ( N.  X.  N. ) )
21breqi 3935 . 2  |-  ( A 
<N  B  <->  A (  _E  i^i  ( N.  X.  N. )
) B )
3 brinxp 4607 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  _E  B  <->  A (  _E  i^i  ( N.  X.  N. ) ) B ) )
4 epelg 4212 . . . 4  |-  ( B  e.  N.  ->  ( A  _E  B  <->  A  e.  B ) )
54adantl 275 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  _E  B  <->  A  e.  B ) )
63, 5bitr3d 189 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A (  _E 
i^i  ( N.  X.  N. ) ) B  <->  A  e.  B ) )
72, 6syl5bb 191 1  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1480    i^i cin 3070   class class class wbr 3929    _E cep 4209    X. cxp 4537   N.cnpi 7087    <N clti 7090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-eprel 4211  df-xp 4545  df-lti 7122
This theorem is referenced by:  ltsopi  7135  pitric  7136  pitri3or  7137  ltdcpi  7138  ltexpi  7152  ltapig  7153  ltmpig  7154  1lt2pi  7155  nlt1pig  7156  archnqq  7232  prarloclemarch2  7234  prarloclemlt  7308  prarloclemn  7314
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