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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

Proof of Theorem uni0b
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsn 3829 . . 3
21ralbii 2729 . 2
3 dfss3 3338 . 2
4 neq0 3638 . . . 4
5 rexcom4 2975 . . . . 5
6 neq0 3638 . . . . . 6
76rexbii 2730 . . . . 5
8 eluni2 4019 . . . . . 6
98exbii 1592 . . . . 5
105, 7, 93bitr4ri 270 . . . 4
11 rexnal 2716 . . . 4
124, 10, 113bitri 263 . . 3
1312con4bii 289 . 2
142, 3, 133bitr4ri 270 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 177  wex 1550   wceq 1652   wcel 1725  wral 2705  wrex 2706   wss 3320  c0 3628  csn 3814  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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