Theorem unidif0 5335

Description: The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)

Assertion

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

Proof of Theorem unidif0

Step	Hyp	Ref	Expression
1		uniun 4911	. . . 4 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
2		undif1 4456	. . . . . 6 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
3		uncom 4138	. . . . . 6 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
4	2, 3	eqtr2i 2760	. . . . 5 ⊢ ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
5	4	unieqi 4900	. . . 4 ⊢ ∪ ({∅} ∪ 𝐴) = ∪ ((𝐴 ∖ {∅}) ∪ {∅})
6		0ex 5282	. . . . . . 7 ⊢ ∅ ∈ V
7	6	unisn 4907	. . . . . 6 ⊢ ∪ {∅} = ∅
8	7	uneq2i 4145	. . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅)
9		un0 4374	. . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅})
10	8, 9	eqtr2i 2760	. . . 4 ⊢ ∪ (𝐴 ∖ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
11	1, 5, 10	3eqtr4ri 2770	. . 3 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ ({∅} ∪ 𝐴)
12		uniun 4911	. . 3 ⊢ ∪ ({∅} ∪ 𝐴) = (∪ {∅} ∪ ∪ 𝐴)
13	7	uneq1i 4144	. . 3 ⊢ (∪ {∅} ∪ ∪ 𝐴) = (∅ ∪ ∪ 𝐴)
14	11, 12, 13	3eqtri 2763	. 2 ⊢ ∪ (𝐴 ∖ {∅}) = (∅ ∪ ∪ 𝐴)
15		uncom 4138	. 2 ⊢ (∅ ∪ ∪ 𝐴) = (∪ 𝐴 ∪ ∅)
16		un0 4374	. 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
17	14, 15, 16	3eqtri 2763	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:   = wceq 1540   ∖ cdif 3928   ∪ cun 3929  ∅c0 4313  {csn 4606  ∪ cuni 4888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-sn 4607  df-pr 4609  df-uni 4889

This theorem is referenced by:  infeq5i  9655  zornn0g  10524  basdif0  22896  tgdif0  22935  omsmeas  34360  stoweidlem57  46066