Theorem unidif0 5330

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 4437	. . . 4 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
2	1	unieqi 4884	. . 3 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = ∪ (𝐴 ∪ {∅})
3		uniun 4895	. . 3 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
4		0ex 5270	. . . . 5 ⊢ ∅ ∈ V
5	4	unisn 4891	. . . 4 ⊢ ∪ {∅} = ∅
6	5	uneq2i 4119	. . 3 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
7	2, 3, 6	3eqtri 2790	. 2 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ 𝐴 ∪ ∅)
8		uniun 4895	. . 3 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
9	5	uneq2i 4119	. . 3 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅)
10		un0 4351	. . 3 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅})
11	8, 9, 10	3eqtri 2790	. 2 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = ∪ (𝐴 ∖ {∅})
12		un0 4351	. 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
13	7, 11, 12	3eqtr3i 2794	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:   = wceq 1570   ∖ cdif 3902   ∪ cun 3903  ∅c0 4286  {csn 4589  ∪ cuni 4872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-sn 4590  df-pr 4592  df-uni 4873

This theorem is referenced by:  infeq5i  9601  zornn0g  10484  basdif0  23110  tgdif0  23149  omsmeas  34713  stoweidlem57  46771