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Theorem uni0b 3852
Description: The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)
Assertion
Ref Expression
uni0b  |-  ( U. A  =  (/)  <->  A  C_  { (/) } )

Proof of Theorem uni0b
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsn 3655 . . 3  |-  ( x  e.  { (/) }  <->  x  =  (/) )
21ralbii 2567 . 2  |-  ( A. x  e.  A  x  e.  { (/) }  <->  A. x  e.  A  x  =  (/) )
3 dfss3 3170 . 2  |-  ( A 
C_  { (/) }  <->  A. x  e.  A  x  e.  {
(/) } )
4 neq0 3465 . . . 4  |-  ( -. 
U. A  =  (/)  <->  E. y  y  e.  U. A
)
5 rexcom4 2807 . . . . 5  |-  ( E. x  e.  A  E. y  y  e.  x  <->  E. y E. x  e.  A  y  e.  x
)
6 neq0 3465 . . . . . 6  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
76rexbii 2568 . . . . 5  |-  ( E. x  e.  A  -.  x  =  (/)  <->  E. x  e.  A  E. y 
y  e.  x )
8 eluni2 3831 . . . . . 6  |-  ( y  e.  U. A  <->  E. x  e.  A  y  e.  x )
98exbii 1569 . . . . 5  |-  ( E. y  y  e.  U. A 
<->  E. y E. x  e.  A  y  e.  x )
105, 7, 93bitr4ri 269 . . . 4  |-  ( E. y  y  e.  U. A 
<->  E. x  e.  A  -.  x  =  (/) )
11 rexnal 2554 . . . 4  |-  ( E. x  e.  A  -.  x  =  (/)  <->  -.  A. x  e.  A  x  =  (/) )
124, 10, 113bitri 262 . . 3  |-  ( -. 
U. A  =  (/)  <->  -.  A. x  e.  A  x  =  (/) )
1312con4bii 288 . 2  |-  ( U. A  =  (/)  <->  A. x  e.  A  x  =  (/) )
142, 3, 133bitr4ri 269 1  |-  ( U. A  =  (/)  <->  A  C_  { (/) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827
This theorem is referenced by:  uni0c  3853  uni0  3854  fin1a2lem11  8036  zornn0g  8132  0top  16721  filcon  17578  alexsubALTlem2  17742  ordcmp  24886  imfstnrelc  25081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-uni 3828
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