Theorem unidif0 5301

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 4416	. . . 4 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
2	1	unieqi 4862	. . 3 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = ∪ (𝐴 ∪ {∅})
3		uniun 4873	. . 3 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
4		0ex 5242	. . . . 5 ⊢ ∅ ∈ V
5	4	unisn 4869	. . . 4 ⊢ ∪ {∅} = ∅
6	5	uneq2i 4105	. . 3 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
7	2, 3, 6	3eqtri 2763	. 2 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ 𝐴 ∪ ∅)
8		uniun 4873	. . 3 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
9	5	uneq2i 4105	. . 3 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅)
10		un0 4334	. . 3 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅})
11	8, 9, 10	3eqtri 2763	. 2 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = ∪ (𝐴 ∖ {∅})
12		un0 4334	. 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
13	7, 11, 12	3eqtr3i 2767	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:   = wceq 1542   ∖ cdif 3886   ∪ cun 3887  ∅c0 4273  {csn 4567  ∪ cuni 4850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-sn 4568  df-pr 4570  df-uni 4851

This theorem is referenced by:  infeq5i  9557  zornn0g  10427  basdif0  22918  tgdif0  22957  omsmeas  34467  stoweidlem57  46485