Theorem uni0c 4890

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 4889	. 2 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
2		dfss3 3922	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
3		velsn 4596	. . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4	3	ralbii 3082	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
5	1, 2, 4	3bitri 297	1 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ⊆ wss 3901  ∅c0 4285  {csn 4580  ∪ cuni 4863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-v 3442  df-dif 3904  df-ss 3918  df-nul 4286  df-sn 4581  df-uni 4864

This theorem is referenced by:  fin1a2lem13  10322  fctop  22948  cctop  22950  ssdifidllem  33537  ssmxidllem  33554