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Theorem poinxp 4409
 Description: Intersection of partial order with cross 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 481 . . . . . . . 8
2 brinxp 4408 . . . . . . . 8
31, 1, 2syl2anc 391 . . . . . . 7
43notbid 592 . . . . . 6
5 brinxp 4408 . . . . . . . . 9
65adantr 261 . . . . . . . 8
7 brinxp 4408 . . . . . . . . 9
87adantll 445 . . . . . . . 8
96, 8anbi12d 442 . . . . . . 7
10 brinxp 4408 . . . . . . . 8
1110adantlr 446 . . . . . . 7
129, 11imbi12d 223 . . . . . 6
134, 12anbi12d 442 . . . . 5
1413ralbidva 2322 . . . 4
1514ralbidva 2322 . . 3
1615ralbiia 2338 . 2
17 df-po 4033 . 2
18 df-po 4033 . 2
1916, 17, 183bitr4i 201 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 97   wb 98   wcel 1393  wral 2306   cin 2916   class class class wbr 3764   wpo 4031   cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-po 4033  df-xp 4351 This theorem is referenced by:  soinxp  4410
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