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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.
StepHypRef Expression
1 simpll 753 . . . . . . . 8
2 brinxp 5067 . . . . . . . 8
31, 1, 2syl2anc 661 . . . . . . 7
43notbid 294 . . . . . 6
5 brinxp 5067 . . . . . . . . 9
65adantr 465 . . . . . . . 8
7 brinxp 5067 . . . . . . . . 9
87adantll 713 . . . . . . . 8
96, 8anbi12d 710 . . . . . . 7
10 brinxp 5067 . . . . . . . 8
1110adantlr 714 . . . . . . 7
129, 11imbi12d 320 . . . . . 6
134, 12anbi12d 710 . . . . 5
1413ralbidva 2893 . . . 4
1514ralbidva 2893 . . 3
1615ralbiia 2887 . 2
17 df-po 4805 . 2
18 df-po 4805 . 2
1916, 17, 183bitr4i 277 1
 Colors of variables: wff setvar class 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
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