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Theorem uni0c 4400
 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 4399 . 2 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
2 dfss3 3558 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
3 velsn 4141 . . 3 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43ralbii 2963 . 2 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
51, 2, 43bitri 285 1 ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-uni 4373 This theorem is referenced by:  fin1a2lem13  9117  fctop  20618  cctop  20620
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