Theorem unixpid 5670

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 5128	. . . 4 |- ( A = (/) -> ( A X. A ) = ( (/) X. A ) )
2		0xp 5199	. . . 4 |- ( (/) X. A ) = (/)
3	1, 2	syl6eq 2672	. . 3 |- ( A = (/) -> ( A X. A ) = (/) )
4		unieq 4444	. . . . 5 |- ( ( A X. A ) = (/) -> U. ( A X. A ) = U. (/) )
5	4	unieqd 4446	. . . 4 |- ( ( A X. A ) = (/) -> U. U. ( A X. A ) = U. U. (/) )
6		uni0 4465	. . . . . 6 |- U. (/) = (/)
7	6	unieqi 4445	. . . . 5 |- U. U. (/) = U. (/)
8	7, 6	eqtri 2644	. . . 4 |- U. U. (/) = (/)
9		eqtr 2641	. . . . 5 |- ( ( U. U. ( A X. A ) = U. U. (/) /\ U. U. (/) = (/) ) -> U. U. ( A X. A ) = (/) )
10		eqtr 2641	. . . . . . 7 |- ( ( U. U. ( A X. A ) = (/) /\ (/) = A ) -> U. U. ( A X. A ) = A )
11	10	expcom 451	. . . . . 6 |- ( (/) = A -> ( U. U. ( A X. A ) = (/) -> U. U. ( A X. A ) = A ) )
12	11	eqcoms 2630	. . . . 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 694	. . 3 |- ( ( A X. A ) = (/) -> ( A = (/) -> U. U. ( A X. A ) = A ) )
15	3, 14	mpcom 38	. 2 |- ( A = (/) -> U. U. ( A X. A ) = A )
16		df-ne 2795	. . 3 |- ( A =/= (/) <-> -. A = (/) )
17		xpnz 5553	. . . 4 |- ( ( A =/= (/) /\ A =/= (/) ) <-> ( A X. A ) =/= (/) )
18		unixp 5668	. . . . 5 |- ( ( A X. A ) =/= (/) -> U. U. ( A X. A ) = ( A u. A ) )
19		unidm 3756	. . . . 5 |- ( A u. A ) = A
20	18, 19	syl6eq 2672	. . . 4 |- ( ( A X. A ) =/= (/) -> U. U. ( A X. A ) = A )
21	17, 20	sylbi 207	. . 3 |- ( ( A =/= (/) /\ A =/= (/) ) -> U. U. ( A X. A ) = A )
22	16, 16, 21	sylancbr 700	. 2 |- ( -. A = (/) -> U. U. ( A X. A ) = A )
23	15, 22	pm2.61i 176	1 |- U. U. ( A X. A ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   u. cun 3572   (/)c0 3915   U.cuni 4436   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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  psss  17214