Theorem unidif0 5293

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 4877	. . . 4 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
2		undif1 4421	. . . . . 6 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
3		uncom 4103	. . . . . 6 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
4	2, 3	eqtr2i 2755	. . . . 5 ⊢ ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
5	4	unieqi 4866	. . . 4 ⊢ ∪ ({∅} ∪ 𝐴) = ∪ ((𝐴 ∖ {∅}) ∪ {∅})
6		0ex 5240	. . . . . . 7 ⊢ ∅ ∈ V
7	6	unisn 4873	. . . . . 6 ⊢ ∪ {∅} = ∅
8	7	uneq2i 4110	. . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅)
9		un0 4339	. . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅})
10	8, 9	eqtr2i 2755	. . . 4 ⊢ ∪ (𝐴 ∖ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
11	1, 5, 10	3eqtr4ri 2765	. . 3 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ ({∅} ∪ 𝐴)
12		uniun 4877	. . 3 ⊢ ∪ ({∅} ∪ 𝐴) = (∪ {∅} ∪ ∪ 𝐴)
13	7	uneq1i 4109	. . 3 ⊢ (∪ {∅} ∪ ∪ 𝐴) = (∅ ∪ ∪ 𝐴)
14	11, 12, 13	3eqtri 2758	. 2 ⊢ ∪ (𝐴 ∖ {∅}) = (∅ ∪ ∪ 𝐴)
15		uncom 4103	. 2 ⊢ (∅ ∪ ∪ 𝐴) = (∪ 𝐴 ∪ ∅)
16		un0 4339	. 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
17	14, 15, 16	3eqtri 2758	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:   = wceq 1541   ∖ cdif 3894   ∪ cun 3895  ∅c0 4278  {csn 4571  ∪ cuni 4854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5239

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-sn 4572  df-pr 4574  df-uni 4855

This theorem is referenced by:  infeq5i  9521  zornn0g  10391  basdif0  22863  tgdif0  22902  omsmeas  34328  stoweidlem57  46095