Theorem uni0c 3959

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 3958	. 2 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
2		dfss3 3236	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
3		velsn 3725	. . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4	3	ralbii 2556	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
5	1, 2, 4	3bitri 206	1 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)

Syntax hints:   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  ∅c0 3520  {csn 3708  ∪ cuni 3933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3714  df-uni 3934

This theorem is referenced by: (None)