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Theorem poinxp 4355
Description: Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
poinxp  R  Po  R  i^i  X.  Po

Proof of Theorem poinxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 481 . . . . . . . 8
2 brinxp 4354 . . . . . . . 8  R  R  i^i  X.
31, 1, 2syl2anc 391 . . . . . . 7  R  R  i^i  X.
43notbid 592 . . . . . 6  R  R  i^i  X.
5 brinxp 4354 . . . . . . . . 9  R  R  i^i  X.
65adantr 261 . . . . . . . 8  R  R  i^i  X.
7 brinxp 4354 . . . . . . . . 9  R  R  i^i  X.
87adantll 445 . . . . . . . 8  R  R  i^i  X.
96, 8anbi12d 442 . . . . . . 7  R  R  R  i^i  X.  R  i^i  X.
10 brinxp 4354 . . . . . . . 8  R  R  i^i  X.
1110adantlr 446 . . . . . . 7  R  R  i^i  X.
129, 11imbi12d 223 . . . . . 6  R  R  R  R  i^i  X.  R  i^i  X.  R  i^i  X.
134, 12anbi12d 442 . . . . 5  R  R  R  R  R  i^i  X.  R  i^i  X.  R  i^i  X.  R  i^i  X.
1413ralbidva 2319 . . . 4  R  R  R  R  R  i^i  X.  R  i^i  X.  R  i^i  X.  R  i^i  X.
1514ralbidva 2319 . . 3  R  R  R  R  R  i^i  X.  R  i^i  X.  R  i^i  X.  R  i^i  X.
1615ralbiia 2335 . 2  R  R  R  R  R  i^i  X.  R  i^i  X.  R  i^i  X.  R  i^i  X.
17 df-po 4027 . 2  R  Po  R  R  R  R
18 df-po 4027 . 2  R  i^i  X.  Po  R  i^i  X.  R  i^i  X.  R  i^i  X.  R  i^i  X.
1916, 17, 183bitr4i 201 1  R  Po  R  i^i  X.  Po
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wcel 1393  wral 2303    i^i cin 2913   class class class wbr 3758    Po wpo 4025    X. cxp 4289
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 3869  ax-pow 3921  ax-pr 3938
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 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-br 3759  df-opab 3813  df-po 4027  df-xp 4297
This theorem is referenced by:  soinxp  4356
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