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Theorem unisn2 4501
Description: A version of unisn 4178 without the  A  e.  _V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
Assertion
Ref Expression
unisn2  |-  U. { A }  e.  { (/) ,  A }

Proof of Theorem unisn2
StepHypRef Expression
1 unisng 4179 . . 3  |-  ( A  e.  _V  ->  U. { A }  =  A
)
2 prid2g 4051 . . 3  |-  ( A  e.  _V  ->  A  e.  { (/) ,  A }
)
31, 2eqeltrd 2470 . 2  |-  ( A  e.  _V  ->  U. { A }  e.  { (/) ,  A } )
4 snprc 4007 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
54biimpi 194 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
65unieqd 4173 . . 3  |-  ( -.  A  e.  _V  ->  U. { A }  =  U. (/) )
7 uni0 4190 . . . 4  |-  U. (/)  =  (/)
8 0ex 4497 . . . . 5  |-  (/)  e.  _V
98prid1 4052 . . . 4  |-  (/)  e.  { (/)
,  A }
107, 9eqeltri 2466 . . 3  |-  U. (/)  e.  { (/)
,  A }
116, 10syl6eqel 2478 . 2  |-  ( -.  A  e.  _V  ->  U. { A }  e.  {
(/) ,  A }
)
123, 11pm2.61i 164 1  |-  U. { A }  e.  { (/) ,  A }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1399    e. wcel 1826   _Vcvv 3034   (/)c0 3711   {csn 3944   {cpr 3946   U.cuni 4163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-nul 4496
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-sn 3945  df-pr 3947  df-uni 4164
This theorem is referenced by: (None)
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