Theorem unidif0 3476

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 3196	. . . 4 |- U.((A \ {(/)}) u. {(/)}) = (U.(A \ {(/)}) u. U.{(/)})
2		undif1 2949	. . . . . 6 |- ((A \ {(/)}) u. {(/)}) = (A u. {(/)})
3		uncom 2744	. . . . . 6 |- (A u. {(/)}) = ({(/)} u. A)
4	2, 3	eqtr2i 1909	. . . . 5 |- ({(/)} u. A) = ((A \ {(/)}) u. {(/)})
5	4	unieqi 3187	. . . 4 |- U.({(/)} u. A) = U.((A \ {(/)}) u. {(/)})
6		0ex 3446	. . . . . . 7 |- (/) e. _V
7	6	unisn 3193	. . . . . 6 |- U.{(/)} = (/)
8	7	uneq2i 2752	. . . . 5 |- (U.(A \ {(/)}) u. U.{(/)}) = (U.(A \ {(/)}) u. (/))
9		un0 2896	. . . . 5 |- (U.(A \ {(/)}) u. (/)) = U.(A \ {(/)})
10	8, 9	eqtr2i 1909	. . . 4 |- U.(A \ {(/)}) = (U.(A \ {(/)}) u. U.{(/)})
11	1, 5, 10	3eqtr4ri 1923	. . 3 |- U.(A \ {(/)}) = U.({(/)} u. A)
12		uniun 3196	. . 3 |- U.({(/)} u. A) = (U.{(/)} u. U.A)
13	7	uneq1i 2751	. . 3 |- (U.{(/)} u. U.A) = ((/) u. U.A)
14	11, 12, 13	3eqtri 1912	. 2 |- U.(A \ {(/)}) = ((/) u. U.A)
15		uncom 2744	. 2 |- ((/) u. U.A) = (U.A u. (/))
16		un0 2896	. 2 |- (U.A u. (/)) = U.A
17	14, 15, 16	3eqtri 1912	1 |- U.(A \ {(/)}) = U.A

Syntax hints:   = wceq 1298   \ cdif 2590   u. cun 2591  (/)c0 2875  {csn 3044  U.cuni 3177

This theorem is referenced by:  infeq5 5727  zornn0 15764

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-nul 3445

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-sn 3049  df-pr 3050  df-uni 3178

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