Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  uni0b Unicode version

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.
StepHypRef Expression
1 elsn 4043 . . 3
21ralbii 2888 . 2
3 dfss3 3493 . 2
4 neq0 3795 . . . 4
5 rexcom4 3129 . . . . 5
6 neq0 3795 . . . . . 6
76rexbii 2959 . . . . 5
8 eluni2 4253 . . . . . 6
98exbii 1667 . . . . 5
105, 7, 93bitr4ri 278 . . . 4
11 rexnal 2905 . . . 4
124, 10, 113bitri 271 . . 3
1312con4bii 297 . 2
142, 3, 133bitr4ri 278 1
 Colors of variables: wff setvar class 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
 Copyright terms: Public domain W3C validator