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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
StepHypRef Expression
1 xpeq1 4965 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  A
) )
2 0xp 5028 . . . 4  |-  ( (/)  X.  A )  =  (/)
31, 2syl6eq 2511 . . 3  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
4 unieq 4210 . . . . 5  |-  ( ( A  X.  A )  =  (/)  ->  U. ( A  X.  A )  = 
U. (/) )
54unieqd 4212 . . . 4  |-  ( ( A  X.  A )  =  (/)  ->  U. U. ( A  X.  A
)  =  U. U. (/) )
6 uni0 4229 . . . . . 6  |-  U. (/)  =  (/)
76unieqi 4211 . . . . 5  |-  U. U. (/)  =  U. (/)
87, 6eqtri 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 )
1110expcom 435 . . . . . 6  |-  ( (/)  =  A  ->  ( U. U. ( A  X.  A
)  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
1211eqcoms 2466 . . . . 5  |-  ( A  =  (/)  ->  ( U. U. ( A  X.  A
)  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
139, 12syl5com 30 . . . 4  |-  ( ( U. U. ( A  X.  A )  = 
U. U. (/)  /\  U. U. (/)  =  (/) )  ->  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
145, 8, 13sylancl 662 . . 3  |-  ( ( A  X.  A )  =  (/)  ->  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
153, 14mpcom 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
2018, 19syl6eq 2511 . . . 4  |-  ( ( A  X.  A )  =/=  (/)  ->  U. U. ( A  X.  A )  =  A )
2117, 20sylbi 195 . . 3  |-  ( ( A  =/=  (/)  /\  A  =/=  (/) )  ->  U. U. ( A  X.  A
)  =  A )
2216, 16, 21sylancbr 666 . 2  |-  ( -.  A  =  (/)  ->  U. U. ( A  X.  A
)  =  A )
2315, 22pm2.61i 164 1  |-  U. U. ( A  X.  A
)  =  A
Colors of variables: wff setvar class
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
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