Theorem ltpiord 9317

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 9305	. . 3 |- <N = ( _E i^i ( N. X. N. ) )
2	1	breqi 4411	. 2 |- ( A <N B <-> A ( _E i^i ( N. X. N. ) ) B )
3		brinxp 4900	. . 3 |- ( ( A e. N. /\ B e. N. ) -> ( A _E B <-> A ( _E i^i ( N. X. N. ) ) B ) )
4		epelg 4749	. . . 4 |- ( B e. N. -> ( A _E B <-> A e. B ) )
5	4	adantl 468	. . 3 |- ( ( A e. N. /\ B e. N. ) -> ( A _E B <-> A e. B ) )
6	3, 5	bitr3d 259	. 2 |- ( ( A e. N. /\ B e. N. ) -> ( A ( _E i^i ( N. X. N. ) ) B <-> A e. B ) )
7	2, 6	syl5bb 261	1 |- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> A e. B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   e. wcel 1889   i^i cin 3405   class class class wbr 4405   _E cep 4746   X. cxp 4835   N.cnpi 9274   <N clti 9277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-eprel 4748  df-xp 4843  df-lti 9305

This theorem is referenced by:  ltexpi  9332  ltapi  9333  ltmpi  9334  1lt2pi  9335  nlt1pi  9336  indpi  9337  nqereu  9359