Theorem ltpiord 7434

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

Step	Hyp	Ref	Expression
1		df-lti 7422	. . 3 |- <N = ( _E i^i ( N. X. N. ) )
2	1	breqi 4051	. 2 |- ( A <N B <-> A ( _E i^i ( N. X. N. ) ) B )
3		brinxp 4744	. . 3 |- ( ( A e. N. /\ B e. N. ) -> ( A _E B <-> A ( _E i^i ( N. X. N. ) ) B ) )
4		epelg 4338	. . . 4 |- ( B e. N. -> ( A _E B <-> A e. B ) )
5	4	adantl 277	. . 3 |- ( ( A e. N. /\ B e. N. ) -> ( A _E B <-> A e. B ) )
6	3, 5	bitr3d 190	. 2 |- ( ( A e. N. /\ B e. N. ) -> ( A ( _E i^i ( N. X. N. ) ) B <-> A e. B ) )
7	2, 6	bitrid 192	1 |- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> A e. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2176   i^i cin 3165   class class class wbr 4045   _E cep 4335   X. cxp 4674   N.cnpi 7387   <N clti 7390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-eprel 4337  df-xp 4682  df-lti 7422

This theorem is referenced by:  ltsopi  7435  pitric  7436  pitri3or  7437  ltdcpi  7438  ltexpi  7452  ltapig  7453  ltmpig  7454  1lt2pi  7455  nlt1pig  7456  archnqq  7532  prarloclemarch2  7534  prarloclemlt  7608  prarloclemn  7614