Theorem ltpiord 9709

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 9697	. . 3 |- <N = ( _E i^i ( N. X. N. ) )
2	1	breqi 4659	. 2 |- ( A <N B <-> A ( _E i^i ( N. X. N. ) ) B )
3		brinxp 5181	. . 3 |- ( ( A e. N. /\ B e. N. ) -> ( A _E B <-> A ( _E i^i ( N. X. N. ) ) B ) )
4		epelg 5030	. . . 4 |- ( B e. N. -> ( A _E B <-> A e. B ) )
5	4	adantl 482	. . 3 |- ( ( A e. N. /\ B e. N. ) -> ( A _E B <-> A e. B ) )
6	3, 5	bitr3d 270	. 2 |- ( ( A e. N. /\ B e. N. ) -> ( A ( _E i^i ( N. X. N. ) ) B <-> A e. B ) )
7	2, 6	syl5bb 272	1 |- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> A e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   i^i cin 3573   class class class wbr 4653   _E cep 5028   X. cxp 5112   N.cnpi 9666   <N clti 9669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-xp 5120  df-lti 9697

This theorem is referenced by:  ltexpi  9724  ltapi  9725  ltmpi  9726  1lt2pi  9727  nlt1pi  9728  indpi  9729  nqereu  9751