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Theorem unisn2 4425
Description: A version of unisn 4103 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 4104 . . 3  |-  ( A  e.  _V  ->  U. { A }  =  A
)
2 prid2g 3979 . . 3  |-  ( A  e.  _V  ->  A  e.  { (/) ,  A }
)
31, 2eqeltrd 2515 . 2  |-  ( A  e.  _V  ->  U. { A }  e.  { (/) ,  A } )
4 snprc 3936 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
54biimpi 194 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
65unieqd 4098 . . 3  |-  ( -.  A  e.  _V  ->  U. { A }  =  U. (/) )
7 uni0 4115 . . . 4  |-  U. (/)  =  (/)
8 0ex 4419 . . . . 5  |-  (/)  e.  _V
98prid1 3980 . . . 4  |-  (/)  e.  { (/)
,  A }
107, 9eqeltri 2511 . . 3  |-  U. (/)  e.  { (/)
,  A }
116, 10syl6eqel 2529 . 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 1364    e. wcel 1761   _Vcvv 2970   (/)c0 3634   {csn 3874   {cpr 3876   U.cuni 4088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-nul 4418
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-sn 3875  df-pr 3877  df-uni 4089
This theorem is referenced by: (None)
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