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Theorem uni0c 4851
 Description: The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
uni0c ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem uni0c
StepHypRef Expression
1 uni0b 4850 . 2 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
2 dfss3 3941 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
3 velsn 4566 . . 3 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43ralbii 3160 . 2 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
51, 2, 43bitri 300 1 ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2115  ∀wral 3133   ⊆ wss 3919  ∅c0 4276  {csn 4550  ∪ cuni 4824 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-uni 4825 This theorem is referenced by:  fin1a2lem13  9832  fctop  21615  cctop  21617  ssmxidllem  31022
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