Theorem unidif0 2739

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 2519	. . . 4 |- U.((A \ {(/)}) u. {(/)}) = (U.(A \ {(/)}) u. U.{(/)})
2		undif1 2340	. . . . . 6 |- ((A \ {(/)}) u. {(/)}) = (A u. {(/)})
3		uncom 2176	. . . . . 6 |- (A u. {(/)}) = ({(/)} u. A)
4	2, 3	eqtr2 1496	. . . . 5 |- ({(/)} u. A) = ((A \ {(/)}) u. {(/)})
5	4	unieqi 2511	. . . 4 |- U.({(/)} u. A) = U.((A \ {(/)}) u. {(/)})
6		0ex 2711	. . . . . . 7 |- (/) e. V
7	6	unisn 2517	. . . . . 6 |- U.{(/)} = (/)
8	7	uneq2i 2181	. . . . 5 |- (U.(A \ {(/)}) u. U.{(/)}) = (U.(A \ {(/)}) u. (/))
9		un0 2297	. . . . 5 |- (U.(A \ {(/)}) u. (/)) = U.(A \ {(/)})
10	8, 9	eqtr2 1496	. . . 4 |- U.(A \ {(/)}) = (U.(A \ {(/)}) u. U.{(/)})
11	1, 5, 10	3eqtr4r 1506	. . 3 |- U.(A \ {(/)}) = U.({(/)} u. A)
12		uniun 2519	. . 3 |- U.({(/)} u. A) = (U.{(/)} u. U.A)
13	7	uneq1i 2180	. . 3 |- (U.{(/)} u. U.A) = ((/) u. U.A)
14	11, 12, 13	3eqtr 1499	. 2 |- U.(A \ {(/)}) = ((/) u. U.A)
15		uncom 2176	. 2 |- ((/) u. U.A) = (U.A u. (/))
16		un0 2297	. 2 |- (U.A u. (/)) = U.A
17	14, 15, 16	3eqtr 1499	1 |- U.(A \ {(/)}) = U.A

Syntax hints:   = wceq 956   \ cdif 2044   u. cun 2045  (/)c0 2280  {csn 2409  U.cuni 2503

This theorem is referenced by:  infeq5 4621

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-nul 2710

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-uni 2504

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