Theorem unixpid 5548

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 5022	. . . 4 |- ( A = (/) -> ( A X. A ) = ( (/) X. A ) )
2		0xp 5089	. . . 4 |- ( (/) X. A ) = (/)
3	1, 2	syl6eq 2514	. . 3 |- ( A = (/) -> ( A X. A ) = (/) )
4		unieq 4259	. . . . 5 |- ( ( A X. A ) = (/) -> U. ( A X. A ) = U. (/) )
5	4	unieqd 4261	. . . 4 |- ( ( A X. A ) = (/) -> U. U. ( A X. A ) = U. U. (/) )
6		uni0 4278	. . . . . 6 |- U. (/) = (/)
7	6	unieqi 4260	. . . . 5 |- U. U. (/) = U. (/)
8	7, 6	eqtri 2486	. . . 4 |- U. U. (/) = (/)
9		eqtr 2483	. . . . 5 |- ( ( U. U. ( A X. A ) = U. U. (/) /\ U. U. (/) = (/) ) -> U. U. ( A X. A ) = (/) )
10		eqtr 2483	. . . . . . 7 |- ( ( U. U. ( A X. A ) = (/) /\ (/) = A ) -> U. U. ( A X. A ) = A )
11	10	expcom 435	. . . . . 6 |- ( (/) = A -> ( U. U. ( A X. A ) = (/) -> U. U. ( A X. A ) = A ) )
12	11	eqcoms 2469	. . . . 5 |- ( A = (/) -> ( U. U. ( A X. A ) = (/) -> U. U. ( A X. A ) = A ) )
13	9, 12	syl5com 30	. . . 4 |- ( ( U. U. ( A X. A ) = U. U. (/) /\ U. U. (/) = (/) ) -> ( A = (/) -> U. U. ( A X. A ) = A ) )
14	5, 8, 13	sylancl 662	. . 3 |- ( ( A X. A ) = (/) -> ( A = (/) -> U. U. ( A X. A ) = A ) )
15	3, 14	mpcom 36	. 2 |- ( A = (/) -> U. U. ( A X. A ) = A )
16		df-ne 2654	. . 3 |- ( A =/= (/) <-> -. A = (/) )
17		xpnz 5433	. . . 4 |- ( ( A =/= (/) /\ A =/= (/) ) <-> ( A X. A ) =/= (/) )
18		unixp 5546	. . . . 5 |- ( ( A X. A ) =/= (/) -> U. U. ( A X. A ) = ( A u. A ) )
19		unidm 3643	. . . . 5 |- ( A u. A ) = A
20	18, 19	syl6eq 2514	. . . 4 |- ( ( A X. A ) =/= (/) -> U. U. ( A X. A ) = A )
21	17, 20	sylbi 195	. . 3 |- ( ( A =/= (/) /\ A =/= (/) ) -> U. U. ( A X. A ) = A )
22	16, 16, 21	sylancbr 666	. 2 |- ( -. A = (/) -> U. U. ( A X. A ) = A )
23	15, 22	pm2.61i 164	1 |- U. U. ( A X. A ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1395   =/= wne 2652   u. cun 3469   (/)c0 3793   U.cuni 4251   X. cxp 5006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019

This theorem is referenced by:  psss  15971