Theorem unidif0 2744

Description: The removal of the empty set from a class does not affect its union.

Assertion

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

Proof of Theorem unidif0

Step	Hyp	Ref	Expression
1		uniun 2523	. . . 4 ⊢ ∪((A ∖ {∅}) ∪ {∅}) = (∪(A ∖ {∅}) ∪ ∪{∅})
2		undif1 2344	. . . . . 6 ⊢ ((A ∖ {∅}) ∪ {∅}) = (A ∪ {∅})
3		uncom 2179	. . . . . 6 ⊢ (A ∪ {∅}) = ({∅} ∪ A)
4	2, 3	eqtr2 1499	. . . . 5 ⊢ ({∅} ∪ A) = ((A ∖ {∅}) ∪ {∅})
5	4	unieqi 2515	. . . 4 ⊢ ∪({∅} ∪ A) = ∪((A ∖ {∅}) ∪ {∅})
6		0ex 2716	. . . . . . 7 ⊢ ∅ ∈ V
7	6	unisn 2521	. . . . . 6 ⊢ ∪{∅} = ∅
8	7	uneq2i 2184	. . . . 5 ⊢ (∪(A ∖ {∅}) ∪ ∪{∅}) = (∪(A ∖ {∅}) ∪ ∅)
9		un0 2301	. . . . 5 ⊢ (∪(A ∖ {∅}) ∪ ∅) = ∪(A ∖ {∅})
10	8, 9	eqtr2 1499	. . . 4 ⊢ ∪(A ∖ {∅}) = (∪(A ∖ {∅}) ∪ ∪{∅})
11	1, 5, 10	3eqtr4r 1509	. . 3 ⊢ ∪(A ∖ {∅}) = ∪({∅} ∪ A)
12		uniun 2523	. . 3 ⊢ ∪({∅} ∪ A) = (∪{∅} ∪ ∪A)
13	7	uneq1i 2183	. . 3 ⊢ (∪{∅} ∪ ∪A) = (∅ ∪ ∪A)
14	11, 12, 13	3eqtr 1502	. 2 ⊢ ∪(A ∖ {∅}) = (∅ ∪ ∪A)
15		uncom 2179	. 2 ⊢ (∅ ∪ ∪A) = (∪A ∪ ∅)
16		un0 2301	. 2 ⊢ (∪A ∪ ∅) = ∪A
17	14, 15, 16	3eqtr 1502	1 ⊢ ∪(A ∖ {∅}) = ∪A

Syntax hints:   = wceq 958   ∖ cdif 2047   ∪ cun 2048  ∅c0 2283  {csn 2413  ∪cuni 2507

This theorem is referenced by:  infeq5 4630

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-nul 2715

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417  df-uni 2508

Copyright terms: Public domain