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Theorem uni0b 4040
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 3829 . . 3  |-  ( x  e.  { (/) }  <->  x  =  (/) )
21ralbii 2729 . 2  |-  ( A. x  e.  A  x  e.  { (/) }  <->  A. x  e.  A  x  =  (/) )
3 dfss3 3338 . 2  |-  ( A 
C_  { (/) }  <->  A. x  e.  A  x  e.  {
(/) } )
4 neq0 3638 . . . 4  |-  ( -. 
U. A  =  (/)  <->  E. y  y  e.  U. A
)
5 rexcom4 2975 . . . . 5  |-  ( E. x  e.  A  E. y  y  e.  x  <->  E. y E. x  e.  A  y  e.  x
)
6 neq0 3638 . . . . . 6  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
76rexbii 2730 . . . . 5  |-  ( E. x  e.  A  -.  x  =  (/)  <->  E. x  e.  A  E. y 
y  e.  x )
8 eluni2 4019 . . . . . 6  |-  ( y  e.  U. A  <->  E. x  e.  A  y  e.  x )
98exbii 1592 . . . . 5  |-  ( E. y  y  e.  U. A 
<->  E. y E. x  e.  A  y  e.  x )
105, 7, 93bitr4ri 270 . . . 4  |-  ( E. y  y  e.  U. A 
<->  E. x  e.  A  -.  x  =  (/) )
11 rexnal 2716 . . . 4  |-  ( E. x  e.  A  -.  x  =  (/)  <->  -.  A. x  e.  A  x  =  (/) )
124, 10, 113bitri 263 . . 3  |-  ( -. 
U. A  =  (/)  <->  -.  A. x  e.  A  x  =  (/) )
1312con4bii 289 . 2  |-  ( U. A  =  (/)  <->  A. x  e.  A  x  =  (/) )
142, 3, 133bitr4ri 270 1  |-  ( U. A  =  (/)  <->  A  C_  { (/) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   (/)c0 3628   {csn 3814   U.cuni 4015
This theorem is referenced by:  uni0c  4041  uni0  4042  fin1a2lem11  8290  zornn0g  8385  0top  17048  filcon  17915  alexsubALTlem2  18079  ordcmp  26197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820  df-uni 4016
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