Theorem uni0c 4868

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 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)

Distinct variable group:   𝑥,𝐴

Proof of Theorem uni0c

Step	Hyp	Ref	Expression
1		uni0b 4867	. 2 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
2		dfss3 3909	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
3		velsn 4577	. . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4	3	ralbii 3092	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
5	1, 2, 4	3bitri 297	1 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  ∅c0 4256  {csn 4561  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-uni 4840

This theorem is referenced by:  fin1a2lem13  10168  fctop  22154  cctop  22156  ssmxidllem  31641