MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltpiord Unicode version

Theorem ltpiord 8511
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 8499 . . 3  |-  <N  =  (  _E  i^i  ( N.  X.  N. ) )
21breqi 4029 . 2  |-  ( A 
<N  B  <->  A (  _E  i^i  ( N.  X.  N. )
) B )
3 brinxp 4752 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  _E  B  <->  A (  _E  i^i  ( N.  X.  N. ) ) B ) )
4 epelg 4306 . . . 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 1684    i^i cin 3151   class class class wbr 4023    _E cep 4303    X. cxp 4687   N.cnpi 8466    <N clti 8469
This theorem is referenced by:  ltexpi  8526  ltapi  8527  ltmpi  8528  1lt2pi  8529  nlt1pi  8530  indpi  8531  nqereu  8553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-xp 4695  df-lti 8499
  Copyright terms: Public domain W3C validator