Theorem unisn0 41306

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 3987	. 2 ⊢ {∅} ⊆ {∅}
2		uni0b 4855	. 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅})
3	1, 2	mpbir 233	1 ⊢ ∪ {∅} = ∅

Syntax hints:   = wceq 1531   ⊆ wss 3934  ∅c0 4289  {csn 4559  ∪ cuni 4830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-v 3495  df-dif 3937  df-in 3941  df-ss 3950  df-nul 4290  df-sn 4560  df-uni 4831

This theorem is referenced by:  founiiun0  41440  prsal  42593