Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  soinxp Structured version   Unicode version

Theorem soinxp 4943
 Description: Intersection of total order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
soinxp

Proof of Theorem soinxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poinxp 4942 . . 3
2 brinxp 4941 . . . . . 6
3 biidd 230 . . . . . 6
4 brinxp 4941 . . . . . . 7
54ancoms 441 . . . . . 6
62, 3, 53orbi123d 1254 . . . . 5
76ralbidva 2722 . . . 4
87ralbiia 2738 . . 3
91, 8anbi12i 680 . 2
10 df-so 4505 . 2
11 df-so 4505 . 2
129, 10, 113bitr4i 270 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   w3o 936   wcel 1726  wral 2706   cin 3320   class class class wbr 4213   wpo 4502   wor 4503   cxp 4877 This theorem is referenced by:  weinxp  4946  ltsopi  8766  cnso  12847  opsrtoslem2  16546 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-po 4504  df-so 4505  df-xp 4885
 Copyright terms: Public domain W3C validator