Theorem unidif0 4625

Description: The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)

Assertion

Ref	Expression
unidif0

Proof of Theorem unidif0

Step	Hyp	Ref	Expression
1		uniun 4268	. . . 4
2		undif1 3903	. . . . . 6
3		uncom 3647	. . . . . 6
4	2, 3	eqtr2i 2487	. . . . 5
5	4	unieqi 4258	. . . 4
6		0ex 4582	. . . . . . 7
7	6	unisn 4264	. . . . . 6
8	7	uneq2i 3654	. . . . 5
9		un0 3810	. . . . 5
10	8, 9	eqtr2i 2487	. . . 4
11	1, 5, 10	3eqtr4ri 2497	. . 3
12		uniun 4268	. . 3
13	7	uneq1i 3653	. . 3
14	11, 12, 13	3eqtri 2490	. 2
15		uncom 3647	. 2
16		un0 3810	. 2
17	14, 15, 16	3eqtri 2490	1

Syntax hints:  =wceq 1395  \cdif 3472  u.cun 3473  c0 3784  {csn 4029  U.cuni 4249

This theorem is referenced by:  infeq5i  8074  zornn0g  8906  basdif0  19454  tgdif0  19494  stoweidlem57  31839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-nul 4581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-pr 4032  df-uni 4250