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Theorem uni0b 4188
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 3958 . . 3  |-  ( x  e.  { (/) }  <->  x  =  (/) )
21ralbii 2813 . 2  |-  ( A. x  e.  A  x  e.  { (/) }  <->  A. x  e.  A  x  =  (/) )
3 dfss3 3407 . 2  |-  ( A 
C_  { (/) }  <->  A. x  e.  A  x  e.  {
(/) } )
4 neq0 3722 . . . 4  |-  ( -. 
U. A  =  (/)  <->  E. y  y  e.  U. A
)
5 rexcom4 3054 . . . . 5  |-  ( E. x  e.  A  E. y  y  e.  x  <->  E. y E. x  e.  A  y  e.  x
)
6 neq0 3722 . . . . . 6  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
76rexbii 2884 . . . . 5  |-  ( E. x  e.  A  -.  x  =  (/)  <->  E. x  e.  A  E. y 
y  e.  x )
8 eluni2 4167 . . . . . 6  |-  ( y  e.  U. A  <->  E. x  e.  A  y  e.  x )
98exbii 1675 . . . . 5  |-  ( E. y  y  e.  U. A 
<->  E. y E. x  e.  A  y  e.  x )
105, 7, 93bitr4ri 278 . . . 4  |-  ( E. y  y  e.  U. A 
<->  E. x  e.  A  -.  x  =  (/) )
11 rexnal 2830 . . . 4  |-  ( E. x  e.  A  -.  x  =  (/)  <->  -.  A. x  e.  A  x  =  (/) )
124, 10, 113bitri 271 . . 3  |-  ( -. 
U. A  =  (/)  <->  -.  A. x  e.  A  x  =  (/) )
1312con4bii 295 . 2  |-  ( U. A  =  (/)  <->  A. x  e.  A  x  =  (/) )
142, 3, 133bitr4ri 278 1  |-  ( U. A  =  (/)  <->  A  C_  { (/) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1399   E.wex 1620    e. wcel 1826   A.wral 2732   E.wrex 2733    C_ wss 3389   (/)c0 3711   {csn 3944   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
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-in 3396  df-ss 3403  df-nul 3712  df-sn 3945  df-uni 4164
This theorem is referenced by:  uni0c  4189  uni0  4190  fin1a2lem11  8703  zornn0g  8798  0top  19570  filcon  20469  alexsubALTlem2  20633  ordcmp  30065
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