Theorem unisn0 43612

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 4002	. 2 ⊢ {∅} ⊆ {∅}
2		uni0b 4933	. 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅})
3	1, 2	mpbir 230	1 ⊢ ∪ {∅} = ∅

Syntax hints:   = wceq 1542   ⊆ wss 3946  ∅c0 4320  {csn 4624  ∪ cuni 4904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3949  df-in 3953  df-ss 3963  df-nul 4321  df-sn 4625  df-uni 4905

This theorem is referenced by:  founiiun0  43759  prsal  44907