Theorem unidif0 5253

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 4851	. . . 4 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
2		undif1 4424	. . . . . 6 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
3		uncom 4129	. . . . . 6 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
4	2, 3	eqtr2i 2845	. . . . 5 ⊢ ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
5	4	unieqi 4841	. . . 4 ⊢ ∪ ({∅} ∪ 𝐴) = ∪ ((𝐴 ∖ {∅}) ∪ {∅})
6		0ex 5204	. . . . . . 7 ⊢ ∅ ∈ V
7	6	unisn 4848	. . . . . 6 ⊢ ∪ {∅} = ∅
8	7	uneq2i 4136	. . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅)
9		un0 4344	. . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅})
10	8, 9	eqtr2i 2845	. . . 4 ⊢ ∪ (𝐴 ∖ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
11	1, 5, 10	3eqtr4ri 2855	. . 3 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ ({∅} ∪ 𝐴)
12		uniun 4851	. . 3 ⊢ ∪ ({∅} ∪ 𝐴) = (∪ {∅} ∪ ∪ 𝐴)
13	7	uneq1i 4135	. . 3 ⊢ (∪ {∅} ∪ ∪ 𝐴) = (∅ ∪ ∪ 𝐴)
14	11, 12, 13	3eqtri 2848	. 2 ⊢ ∪ (𝐴 ∖ {∅}) = (∅ ∪ ∪ 𝐴)
15		uncom 4129	. 2 ⊢ (∅ ∪ ∪ 𝐴) = (∪ 𝐴 ∪ ∅)
16		un0 4344	. 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
17	14, 15, 16	3eqtri 2848	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:   = wceq 1533   ∖ cdif 3933   ∪ cun 3934  ∅c0 4291  {csn 4561  ∪ cuni 4832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-nul 5203

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-sn 4562  df-pr 4564  df-uni 4833

This theorem is referenced by:  infeq5i  9093  zornn0g  9921  basdif0  21555  tgdif0  21594  omsmeas  31576  stoweidlem57  42335