Theorem unidif0 5328

Description: The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.) (Proof shortened by Eric Schmidt, 25-Apr-2026.)

Assertion

Ref	Expression
unidif0	⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Proof of Theorem unidif0

Step	Hyp	Ref	Expression
1		undif1 4433	. . . 4 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
2	1	unieqi 4884	. . 3 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = ∪ (𝐴 ∪ {∅})
3		uniun 4895	. . 3 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
4		0ex 5267	. . . . 5 ⊢ ∅ ∈ V
5	4	unisn 4891	. . . 4 ⊢ ∪ {∅} = ∅
6	5	uneq2i 4167	. . 3 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
7	2, 3, 6	3eqtri 2790	. 2 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ 𝐴 ∪ ∅)
8		uniun 4895	. . 3 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
9	5	uneq2i 4167	. . 3 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅)
10		un0 4347	. . 3 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅})
11	8, 9, 10	3eqtri 2790	. 2 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = ∪ (𝐴 ∖ {∅})
12		un0 4347	. 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
13	7, 11, 12	3eqtr3i 2794	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 4048   ∪ cun 4152  ∅c0 4281  {csn 4588  ∪ cuni 4872

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591  df-uni 4873

This theorem is used by:  infeq5i  9674  zornn0g  10622  basdif0  23326  tgdif0  23365  omsmeas  35066  stoweidlem57  47340