Theorem unisn0 45632

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 3961	. 2 ⊢ {∅} ⊆ {∅}
2		uni0b 4895	. 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅})
3	1, 2	mpbir 234	1 ⊢ ∪ {∅} = ∅

Syntax hints:   = wceq 1563   ⊆ wss 3907  ∅c0 4288  {csn 4585  ∪ cuni 4868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-v 3459  df-dif 3910  df-ss 3924  df-nul 4289  df-sn 4586  df-uni 4869

This theorem is referenced by:  founiiun0  45766  prsal  46890