Theorem uni0b 4274

Description: The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)

Assertion

Ref	Expression
uni0b

Proof of Theorem uni0b

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsn 4043	. . 3
2	1	ralbii 2888	. 2
3		dfss3 3493	. 2
4		neq0 3795	. . . 4
5		rexcom4 3129	. . . . 5
6		neq0 3795	. . . . . 6
7	6	rexbii 2959	. . . . 5
8		eluni2 4253	. . . . . 6
9	8	exbii 1667	. . . . 5
10	5, 7, 9	3bitr4ri 278	. . . 4
11		rexnal 2905	. . . 4
12	4, 10, 11	3bitri 271	. . 3
13	12	con4bii 297	. 2
14	2, 3, 13	3bitr4ri 278	1

Syntax hints:  -.wn 3  <->wb 184  =wceq 1395  E.wex 1612  e.wcel 1818  A.wral 2807  E.wrex 2808  C_wss 3475  c0 3784  {csn 4029  U.cuni 4249

This theorem is referenced by:  uni0c  4275  uni0  4276  fin1a2lem11  8811  zornn0g  8906  0top  19485  filcon  20384  alexsubALTlem2  20548  ordcmp  29912

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

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-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-uni 4250