Theorem ltpiord 7650

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 7638	. . 3 |- <N = ( _E i^i ( N. X. N. ) )
2	1	breqi 4120	. 2 |- ( A <N B <-> A ( _E i^i ( N. X. N. ) ) B )
3		brinxp 4823	. . 3 |- ( ( A e. N. /\ B e. N. ) -> ( A _E B <-> A ( _E i^i ( N. X. N. ) ) B ) )
4		epelg 4416	. . . 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 2205   i^i cin 3213   class class class wbr 4114   _E cep 4413   X. cxp 4752   N.cnpi 7603   <N clti 7606

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-eprel 4415  df-xp 4760  df-lti 7638

This theorem is referenced by:  ltsopi  7651  pitric  7652  pitri3or  7653  ltdcpi  7654  ltexpi  7668  ltapig  7669  ltmpig  7670  1lt2pi  7671  nlt1pig  7672  archnqq  7748  prarloclemarch2  7750  prarloclemlt  7824  prarloclemn  7830