Theorem unidif0 5302

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 4884	. . . 4 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
2		undif1 4429	. . . . . 6 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
3		uncom 4111	. . . . . 6 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
4	2, 3	eqtr2i 2753	. . . . 5 ⊢ ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
5	4	unieqi 4873	. . . 4 ⊢ ∪ ({∅} ∪ 𝐴) = ∪ ((𝐴 ∖ {∅}) ∪ {∅})
6		0ex 5249	. . . . . . 7 ⊢ ∅ ∈ V
7	6	unisn 4880	. . . . . 6 ⊢ ∪ {∅} = ∅
8	7	uneq2i 4118	. . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅)
9		un0 4347	. . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅})
10	8, 9	eqtr2i 2753	. . . 4 ⊢ ∪ (𝐴 ∖ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
11	1, 5, 10	3eqtr4ri 2763	. . 3 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ ({∅} ∪ 𝐴)
12		uniun 4884	. . 3 ⊢ ∪ ({∅} ∪ 𝐴) = (∪ {∅} ∪ ∪ 𝐴)
13	7	uneq1i 4117	. . 3 ⊢ (∪ {∅} ∪ ∪ 𝐴) = (∅ ∪ ∪ 𝐴)
14	11, 12, 13	3eqtri 2756	. 2 ⊢ ∪ (𝐴 ∖ {∅}) = (∅ ∪ ∪ 𝐴)
15		uncom 4111	. 2 ⊢ (∅ ∪ ∪ 𝐴) = (∪ 𝐴 ∪ ∅)
16		un0 4347	. 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
17	14, 15, 16	3eqtri 2756	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:   = wceq 1540   ∖ cdif 3902   ∪ cun 3903  ∅c0 4286  {csn 4579  ∪ cuni 4861

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 2701  ax-nul 5248

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 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-sn 4580  df-pr 4582  df-uni 4862

This theorem is referenced by:  infeq5i  9551  zornn0g  10418  basdif0  22857  tgdif0  22896  omsmeas  34310  stoweidlem57  46058