Theorem unidif0 1944

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

Syntax hints:   = wceq 1091   \ cdif 1484   u. cun 1485  (/)c0 1707  {csn 1808  U.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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