Theorem unisn0 45896

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 3956	. 2 ⊢ {∅} ⊆ {∅}
2		uni0b 4897	. 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅})
3	1, 2	mpbir 234	1 ⊢ ∪ {∅} = ∅

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3902  ∅c0 4282  {csn 4587  ∪ cuni 4870

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-v 3455  df-dif 3905  df-ss 3919  df-nul 4283  df-sn 4588  df-uni 4871

This theorem is used by:  founiiun0  46030  prsal  47154