Theorem unidif0 2729

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

Assertion

Ref	Expression
unidif0	|- U.(A \ {(/)}) = U.A

Proof of Theorem unidif0

Step	Hyp	Ref	Expression
1		uniun 2509	. . . 4 |- U.((A \ {(/)}) u. {(/)}) = (U.(A \ {(/)}) u. U.{(/)})
2		undif1 2330	. . . . . 6 |- ((A \ {(/)}) u. {(/)}) = (A u. {(/)})
3		uncom 2166	. . . . . 6 |- (A u. {(/)}) = ({(/)} u. A)
4	2, 3	eqtr2 1488	. . . . 5 |- ({(/)} u. A) = ((A \ {(/)}) u. {(/)})
5	4	unieqi 2501	. . . 4 |- U.({(/)} u. A) = U.((A \ {(/)}) u. {(/)})
6		0ex 2701	. . . . . . 7 |- (/) e. V
7	6	unisn 2507	. . . . . 6 |- U.{(/)} = (/)
8	7	uneq2i 2171	. . . . 5 |- (U.(A \ {(/)}) u. U.{(/)}) = (U.(A \ {(/)}) u. (/))
9		un0 2287	. . . . 5 |- (U.(A \ {(/)}) u. (/)) = U.(A \ {(/)})
10	8, 9	eqtr2 1488	. . . 4 |- U.(A \ {(/)}) = (U.(A \ {(/)}) u. U.{(/)})
11	1, 5, 10	3eqtr4r 1498	. . 3 |- U.(A \ {(/)}) = U.({(/)} u. A)
12		uniun 2509	. . 3 |- U.({(/)} u. A) = (U.{(/)} u. U.A)
13	7	uneq1i 2170	. . 3 |- (U.{(/)} u. U.A) = ((/) u. U.A)
14	11, 12, 13	3eqtr 1491	. 2 |- U.(A \ {(/)}) = ((/) u. U.A)
15		uncom 2166	. 2 |- ((/) u. U.A) = (U.A u. (/))
16		un0 2287	. 2 |- (U.A u. (/)) = U.A
17	14, 15, 16	3eqtr 1491	1 |- U.(A \ {(/)}) = U.A

Syntax hints:   = wceq 953   \ cdif 2034   u. cun 2035  (/)c0 2270  {csn 2399  U.cuni 2493

This theorem is referenced by:  infeq5 4593

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-nul 2700

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-sn 2402  df-pr 2403  df-uni 2494

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