Theorem unidif0 1944

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 1934	. . . 4 ⊢ ∪((A ∖ {∅}) ∪ {∅}) = (∪(A ∖ {∅}) ∪ ∪{∅})
2		undif1 1761	. . . . . 6 ⊢ ((A ∖ {∅}) ∪ {∅}) = (A ∪ {∅})
3		uncom 1604	. . . . . 6 ⊢ (A ∪ {∅}) = ({∅} ∪ A)
4	2, 3	eqtr2 1120	. . . . 5 ⊢ ({∅} ∪ A) = ((A ∖ {∅}) ∪ {∅})
5	4	unieqi 1928	. . . 4 ⊢ ∪({∅} ∪ A) = ∪((A ∖ {∅}) ∪ {∅})
6		0ex 1745	. . . . . . 7 ⊢ ∅ ∈ V
7	6	unisn 1932	. . . . . 6 ⊢ ∪{∅} = ∅
8	7	uneq2i 1608	. . . . 5 ⊢ (∪(A ∖ {∅}) ∪ ∪{∅}) = (∪(A ∖ {∅}) ∪ ∅)
9		un0 1721	. . . . 5 ⊢ (∪(A ∖ {∅}) ∪ ∅) = ∪(A ∖ {∅})
10	8, 9	eqtr2 1120	. . . 4 ⊢ ∪(A ∖ {∅}) = (∪(A ∖ {∅}) ∪ ∪{∅})
11	1, 5, 10	3eqtr4r 1127	. . 3 ⊢ ∪(A ∖ {∅}) = ∪({∅} ∪ A)
12		uniun 1934	. . 3 ⊢ ∪({∅} ∪ A) = (∪{∅} ∪ ∪A)
13	7	uneq1i 1607	. . 3 ⊢ (∪{∅} ∪ ∪A) = (∅ ∪ ∪A)
14	11, 12, 13	3eqtr 1123	. 2 ⊢ ∪(A ∖ {∅}) = (∅ ∪ ∪A)
15		uncom 1604	. 2 ⊢ (∅ ∪ ∪A) = (∪A ∪ ∅)
16		un0 1721	. 2 ⊢ (∪A ∪ ∅) = ∪A
17	14, 15, 16	3eqtr 1123	1 ⊢ ∪(A ∖ {∅}) = ∪A

Syntax hints:   = wceq 1091   ∖ cdif 1484   ∪ cun 1485  ∅c0 1707  {csn 1808  ∪cuni 1919

This theorem is referenced by:  inf5 3472

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-uni 1920

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