Theorem uni0c 3918

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 3917	. 2 ⊢ (∪A = ∅ ↔ A ⊆ {∅})
2		dfss3 3264	. 2 ⊢ (A ⊆ {∅} ↔ ∀x ∈ A x ∈ {∅})
3		elsn 3749	. . 3 ⊢ (x ∈ {∅} ↔ x = ∅)
4	3	ralbii 2639	. 2 ⊢ (∀x ∈ A x ∈ {∅} ↔ ∀x ∈ A x = ∅)
5	1, 2, 4	3bitri 262	1 ⊢ (∪A = ∅ ↔ ∀x ∈ A x = ∅)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2615   ⊆ wss 3258  ∅c0 3551  {csn 3738  ∪cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-uni 3893

This theorem is referenced by: (None)