Theorem unixpid 5483

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 4965	. . . 4 |- ( A = (/) -> ( A X. A ) = ( (/) X. A ) )
2		0xp 5028	. . . 4 |- ( (/) X. A ) = (/)
3	1, 2	syl6eq 2511	. . 3 |- ( A = (/) -> ( A X. A ) = (/) )
4		unieq 4210	. . . . 5 |- ( ( A X. A ) = (/) -> U. ( A X. A ) = U. (/) )
5	4	unieqd 4212	. . . 4 |- ( ( A X. A ) = (/) -> U. U. ( A X. A ) = U. U. (/) )
6		uni0 4229	. . . . . 6 |- U. (/) = (/)
7	6	unieqi 4211	. . . . 5 |- U. U. (/) = U. (/)
8	7, 6	eqtri 2483	. . . 4 |- U. U. (/) = (/)
9		eqtr 2480	. . . . 5 |- ( ( U. U. ( A X. A ) = U. U. (/) /\ U. U. (/) = (/) ) -> U. U. ( A X. A ) = (/) )
10		eqtr 2480	. . . . . . 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 2466	. . . . 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 2650	. . 3 |- ( A =/= (/) <-> -. A = (/) )
17		xpnz 5368	. . . 4 |- ( ( A =/= (/) /\ A =/= (/) ) <-> ( A X. A ) =/= (/) )
18		unixp 5481	. . . . 5 |- ( ( A X. A ) =/= (/) -> U. U. ( A X. A ) = ( A u. A ) )
19		unidm 3610	. . . . 5 |- ( A u. A ) = A
20	18, 19	syl6eq 2511	. . . 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 1370   =/= wne 2648   u. cun 3437   (/)c0 3748   U.cuni 4202   X. cxp 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-dm 4961  df-rn 4962

This theorem is referenced by:  psss  15507