Theorem unidif0 5331

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 4442	. . . 4 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
2	1	unieqi 4888	. . 3 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = ∪ (𝐴 ∪ {∅})
3		uniun 4899	. . 3 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
4		0ex 5272	. . . . 5 ⊢ ∅ ∈ V
5	4	unisn 4895	. . . 4 ⊢ ∪ {∅} = ∅
6	5	uneq2i 4127	. . 3 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
7	2, 3, 6	3eqtri 2796	. 2 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ 𝐴 ∪ ∅)
8		uniun 4899	. . 3 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
9	5	uneq2i 4127	. . 3 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅)
10		un0 4358	. . 3 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅})
11	8, 9, 10	3eqtri 2796	. 2 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = ∪ (𝐴 ∖ {∅})
12		un0 4358	. 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
13	7, 11, 12	3eqtr3i 2800	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:   = wceq 1567   ∖ cdif 3910   ∪ cun 3911  ∅c0 4294  {csn 4594  ∪ cuni 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-sn 4595  df-pr 4597  df-uni 4877

This theorem is referenced by:  infeq5i  9604  zornn0g  10488  basdif0  23078  tgdif0  23117  omsmeas  34657  stoweidlem57  46662