Theorem unisn0 39981

Description: The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
unisn0	⊢ ∪ {∅} = ∅

Proof of Theorem unisn0

Step	Hyp	Ref	Expression
1		ssid 3819	. 2 ⊢ {∅} ⊆ {∅}
2		uni0b 4655	. 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅})
3	1, 2	mpbir 223	1 ⊢ ∪ {∅} = ∅

Syntax hints:   = wceq 1653   ⊆ wss 3769  ∅c0 4115  {csn 4368  ∪ cuni 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-in 3776  df-ss 3783  df-nul 4116  df-sn 4369  df-uni 4629

This theorem is referenced by:  founiiun0  40131