Theorem ltpiord 9286

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

Proof of Theorem ltpiord

Step	Hyp	Ref	Expression
1		df-lti 9274	. . 3
2	1	breqi 4458	. 2
3		brinxp 5067	. . 3
4		epelg 4797	. . . 4
5	4	adantl 466	. . 3
6	3, 5	bitr3d 255	. 2
7	2, 6	syl5bb 257	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  e.wcel 1818  i^icin 3474   class class class wbr 4452  cep 4794  X.cxp 5002  cnpi 9243  clti 9246

This theorem is referenced by:  ltexpi  9301  ltapi  9302  ltmpi  9303  1lt2pi  9304  nlt1pi  9305  indpi  9306  nqereu  9328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  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 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-eprel 4796  df-xp 5010  df-lti 9274