Theorem unisn0 45408

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 3958	. 2 ⊢ {∅} ⊆ {∅}
2		uni0b 4891	. 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅})
3	1, 2	mpbir 231	1 ⊢ ∪ {∅} = ∅

Syntax hints:   = wceq 1542   ⊆ wss 3903  ∅c0 4287  {csn 4582  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-v 3444  df-dif 3906  df-ss 3920  df-nul 4288  df-sn 4583  df-uni 4866

This theorem is referenced by:  founiiun0  45543  prsal  46670