Theorem ltpiord 9282

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

Step	Hyp	Ref	Expression
1		df-lti 9270	. . 3 |- <N = ( _E i^i ( N. X. N. ) )
2	1	breqi 4462	. 2 |- ( A <N B <-> A ( _E i^i ( N. X. N. ) ) B )
3		brinxp 5071	. . 3 |- ( ( A e. N. /\ B e. N. ) -> ( A _E B <-> A ( _E i^i ( N. X. N. ) ) B ) )
4		epelg 4801	. . . 4 |- ( B e. N. -> ( A _E B <-> A e. B ) )
5	4	adantl 466	. . 3 |- ( ( A e. N. /\ B e. N. ) -> ( A _E B <-> A e. B ) )
6	3, 5	bitr3d 255	. 2 |- ( ( A e. N. /\ B e. N. ) -> ( A ( _E i^i ( N. X. N. ) ) B <-> A e. B ) )
7	2, 6	syl5bb 257	1 |- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> A e. B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1819   i^i cin 3470   class class class wbr 4456   _E cep 4798   X. cxp 5006   N.cnpi 9239   <N clti 9242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-eprel 4800  df-xp 5014  df-lti 9270

This theorem is referenced by:  ltexpi  9297  ltapi  9298  ltmpi  9299  1lt2pi  9300  nlt1pi  9301  indpi  9302  nqereu  9324