MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unisn2 Structured version   Unicode version

Theorem unisn2 4589
Description: A version of unisn 4266 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 4267 . . 3  |-  ( A  e.  _V  ->  U. { A }  =  A
)
2 prid2g 4140 . . 3  |-  ( A  e.  _V  ->  A  e.  { (/) ,  A }
)
31, 2eqeltrd 2555 . 2  |-  ( A  e.  _V  ->  U. { A }  e.  { (/) ,  A } )
4 snprc 4097 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
54biimpi 194 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
65unieqd 4261 . . 3  |-  ( -.  A  e.  _V  ->  U. { A }  =  U. (/) )
7 uni0 4278 . . . 4  |-  U. (/)  =  (/)
8 0ex 4583 . . . . 5  |-  (/)  e.  _V
98prid1 4141 . . . 4  |-  (/)  e.  { (/)
,  A }
107, 9eqeltri 2551 . . 3  |-  U. (/)  e.  { (/)
,  A }
116, 10syl6eqel 2563 . 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 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   {csn 4033   {cpr 4035   U.cuni 4251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-sn 4034  df-pr 4036  df-uni 4252
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator