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Theorem uni0b 4829
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 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})

Proof of Theorem uni0b
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 velsn 4544 . . 3 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
21ralbii 3136 . 2 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
3 dfss3 3906 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
4 neq0 4262 . . . 4 𝐴 = ∅ ↔ ∃𝑦 𝑦 𝐴)
5 rexcom4 3215 . . . . 5 (∃𝑥𝐴𝑦 𝑦𝑥 ↔ ∃𝑦𝑥𝐴 𝑦𝑥)
6 neq0 4262 . . . . . 6 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
76rexbii 3213 . . . . 5 (∃𝑥𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥𝐴𝑦 𝑦𝑥)
8 eluni2 4807 . . . . . 6 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
98exbii 1849 . . . . 5 (∃𝑦 𝑦 𝐴 ↔ ∃𝑦𝑥𝐴 𝑦𝑥)
105, 7, 93bitr4ri 307 . . . 4 (∃𝑦 𝑦 𝐴 ↔ ∃𝑥𝐴 ¬ 𝑥 = ∅)
11 rexnal 3204 . . . 4 (∃𝑥𝐴 ¬ 𝑥 = ∅ ↔ ¬ ∀𝑥𝐴 𝑥 = ∅)
124, 10, 113bitri 300 . . 3 𝐴 = ∅ ↔ ¬ ∀𝑥𝐴 𝑥 = ∅)
1312con4bii 324 . 2 ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
142, 3, 133bitr4ri 307 1 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209   = wceq 1538  wex 1781  wcel 2112  wral 3109  wrex 3110  wss 3884  c0 4246  {csn 4528   cuni 4803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-uni 4804
This theorem is referenced by:  uni0c  4830  uni0  4831  fin1a2lem11  9825  zornn0g  9920  0top  21592  filconn  22492  alexsubALTlem2  22657  ordcmp  33909  unisn0  41681
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