Theorem poinxp 5068

Description: Intersection of partial order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Assertion

Ref	Expression
poinxp

Proof of Theorem poinxp

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 753	. . . . . . . 8
2		brinxp 5067	. . . . . . . 8
3	1, 1, 2	syl2anc 661	. . . . . . 7
4	3	notbid 294	. . . . . 6
5		brinxp 5067	. . . . . . . . 9
6	5	adantr 465	. . . . . . . 8
7		brinxp 5067	. . . . . . . . 9
8	7	adantll 713	. . . . . . . 8
9	6, 8	anbi12d 710	. . . . . . 7
10		brinxp 5067	. . . . . . . 8
11	10	adantlr 714	. . . . . . 7
12	9, 11	imbi12d 320	. . . . . 6
13	4, 12	anbi12d 710	. . . . 5
14	13	ralbidva 2893	. . . 4
15	14	ralbidva 2893	. . 3
16	15	ralbiia 2887	. 2
17		df-po 4805	. 2
18		df-po 4805	. 2
19	16, 17, 18	3bitr4i 277	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  e.wcel 1818  A.wral 2807  i^icin 3474   class class class wbr 4452  Powpo 4803  X.cxp 5002

This theorem is referenced by:  soinxp  5069

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-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-po 4805  df-xp 5010