Theorem unixpid 5371

Description: Field of a square Cartesian product. (Contributed by FL, 10-Oct-2009.)

Assertion

Ref	Expression
unixpid	|- U. U. ( A X. A ) = A

Proof of Theorem unixpid

Step	Hyp	Ref	Expression
1		xpeq1 4848	. . . 4 |- ( A = (/) -> ( A X. A ) = ( (/) X. A ) )
2		0xp 4915	. . . 4 |- ( (/) X. A ) = (/)
3	1, 2	syl6eq 2501	. . 3 |- ( A = (/) -> ( A X. A ) = (/) )
4		unieq 4206	. . . . 5 |- ( ( A X. A ) = (/) -> U. ( A X. A ) = U. (/) )
5	4	unieqd 4208	. . . 4 |- ( ( A X. A ) = (/) -> U. U. ( A X. A ) = U. U. (/) )
6		uni0 4225	. . . . . 6 |- U. (/) = (/)
7	6	unieqi 4207	. . . . 5 |- U. U. (/) = U. (/)
8	7, 6	eqtri 2473	. . . 4 |- U. U. (/) = (/)
9		eqtr 2470	. . . . 5 |- ( ( U. U. ( A X. A ) = U. U. (/) /\ U. U. (/) = (/) ) -> U. U. ( A X. A ) = (/) )
10		eqtr 2470	. . . . . . 7 |- ( ( U. U. ( A X. A ) = (/) /\ (/) = A ) -> U. U. ( A X. A ) = A )
11	10	expcom 437	. . . . . 6 |- ( (/) = A -> ( U. U. ( A X. A ) = (/) -> U. U. ( A X. A ) = A ) )
12	11	eqcoms 2459	. . . . 5 |- ( A = (/) -> ( U. U. ( A X. A ) = (/) -> U. U. ( A X. A ) = A ) )
13	9, 12	syl5com 31	. . . 4 |- ( ( U. U. ( A X. A ) = U. U. (/) /\ U. U. (/) = (/) ) -> ( A = (/) -> U. U. ( A X. A ) = A ) )
14	5, 8, 13	sylancl 668	. . 3 |- ( ( A X. A ) = (/) -> ( A = (/) -> U. U. ( A X. A ) = A ) )
15	3, 14	mpcom 37	. 2 |- ( A = (/) -> U. U. ( A X. A ) = A )
16		df-ne 2624	. . 3 |- ( A =/= (/) <-> -. A = (/) )
17		xpnz 5256	. . . 4 |- ( ( A =/= (/) /\ A =/= (/) ) <-> ( A X. A ) =/= (/) )
18		unixp 5369	. . . . 5 |- ( ( A X. A ) =/= (/) -> U. U. ( A X. A ) = ( A u. A ) )
19		unidm 3577	. . . . 5 |- ( A u. A ) = A
20	18, 19	syl6eq 2501	. . . 4 |- ( ( A X. A ) =/= (/) -> U. U. ( A X. A ) = A )
21	17, 20	sylbi 199	. . 3 |- ( ( A =/= (/) /\ A =/= (/) ) -> U. U. ( A X. A ) = A )
22	16, 16, 21	sylancbr 672	. 2 |- ( -. A = (/) -> U. U. ( A X. A ) = A )
23	15, 22	pm2.61i 168	1 |- U. U. ( A X. A ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1444   =/= wne 2622   u. cun 3402   (/)c0 3731   U.cuni 4198   X. cxp 4832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-cnv 4842  df-dm 4844  df-rn 4845

This theorem is referenced by:  psss  16460