Theorem soinxp 5183

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

Assertion

Ref	Expression
soinxp	|- ( R Or A <-> ( R i^i ( A X. A ) ) Or A )

Proof of Theorem soinxp

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

Step	Hyp	Ref	Expression
1		poinxp 5182	. . 3 |- ( R Po A <-> ( R i^i ( A X. A ) ) Po A )
2		brinxp 5181	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
3		biidd 252	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( x = y <-> x = y ) )
4		brinxp 5181	. . . . . . 7 |- ( ( y e. A /\ x e. A ) -> ( y R x <-> y ( R i^i ( A X. A ) ) x ) )
5	4	ancoms 469	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( y R x <-> y ( R i^i ( A X. A ) ) x ) )
6	2, 3, 5	3orbi123d 1398	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( ( x R y \/ x = y \/ y R x ) <-> ( x ( R i^i ( A X. A ) ) y \/ x = y \/ y ( R i^i ( A X. A ) ) x ) ) )
7	6	ralbidva 2985	. . . 4 |- ( x e. A -> ( A. y e. A ( x R y \/ x = y \/ y R x ) <-> A. y e. A ( x ( R i^i ( A X. A ) ) y \/ x = y \/ y ( R i^i ( A X. A ) ) x ) ) )
8	7	ralbiia 2979	. . 3 |- ( A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) <-> A. x e. A A. y e. A ( x ( R i^i ( A X. A ) ) y \/ x = y \/ y ( R i^i ( A X. A ) ) x ) )
9	1, 8	anbi12i 733	. 2 |- ( ( R Po A /\ A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) <-> ( ( R i^i ( A X. A ) ) Po A /\ A. x e. A A. y e. A ( x ( R i^i ( A X. A ) ) y \/ x = y \/ y ( R i^i ( A X. A ) ) x ) ) )
10		df-so 5036	. 2 |- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) )
11		df-so 5036	. 2 |- ( ( R i^i ( A X. A ) ) Or A <-> ( ( R i^i ( A X. A ) ) Po A /\ A. x e. A A. y e. A ( x ( R i^i ( A X. A ) ) y \/ x = y \/ y ( R i^i ( A X. A ) ) x ) ) )
12	9, 10, 11	3bitr4i 292	1 |- ( R Or A <-> ( R i^i ( A X. A ) ) Or A )

Syntax hints:   <-> wb 196   /\ wa 384   \/ w3o 1036   e. wcel 1990   A.wral 2912   i^i cin 3573   class class class wbr 4653   Po wpo 5033   Or wor 5034   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-xp 5120

This theorem is referenced by:  weinxp  5186  ltsopi  9710  cnso  14976  opsrtoslem2  19485