Theorem unidif0 5321

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 4429	. . . 4 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
2	1	unieqi 4878	. . 3 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = ∪ (𝐴 ∪ {∅})
3		uniun 4889	. . 3 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
4		0ex 5260	. . . . 5 ⊢ ∅ ∈ V
5	4	unisn 4885	. . . 4 ⊢ ∪ {∅} = ∅
6	5	uneq2i 4163	. . 3 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
7	2, 3, 6	3eqtri 2787	. 2 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ 𝐴 ∪ ∅)
8		uniun 4889	. . 3 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
9	5	uneq2i 4163	. . 3 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅)
10		un0 4343	. . 3 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅})
11	8, 9, 10	3eqtri 2787	. 2 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = ∪ (𝐴 ∖ {∅})
12		un0 4343	. 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
13	7, 11, 12	3eqtr3i 2791	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

This proof depends on syntax axioms:   = wceq 1569   ∖ cdif 4044   ∪ cun 4148  ∅c0 4277  {csn 4584  ∪ cuni 4866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-sn 4585  df-pr 4587  df-uni 4867

This theorem is used by:  infeq5i  9644  zornn0g  10590  basdif0  23280  tgdif0  23319  omsmeas  35005  stoweidlem57  47175