ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uni0b Structured version   Unicode version

Theorem uni0b 3596
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  U.  (/)  C_  {
(/) }

Proof of Theorem uni0b
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eq0 3233 . . . 4  (/)
21ralbii 2324 . . 3  (/)
3 ralcom4 2570 . . 3
42, 3bitri 173 . 2  (/)
5 dfss3 2929 . . 3 
C_  { (/) }  { (/)
}
6 elsn 3382 . . . 4  { (/) }  (/)
76ralbii 2324 . . 3  { (/) }  (/)
85, 7bitri 173 . 2 
C_  { (/) }  (/)
9 eluni2 3575 . . . . 5  U.
109notbii 593 . . . 4  U.
1110albii 1356 . . 3  U.
12 eq0 3233 . . 3  U.  (/)  U.
13 ralnex 2310 . . . 4
1413albii 1356 . . 3
1511, 12, 143bitr4i 201 . 2  U.  (/)
164, 8, 153bitr4ri 202 1  U.  (/)  C_  {
(/) }
Colors of variables: wff set class
Syntax hints:   wn 3   wb 98  wal 1240   wceq 1242   wcel 1390  wral 2300  wrex 2301    C_ wss 2911   (/)c0 3218   {csn 3367   U.cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-uni 3572
This theorem is referenced by:  uni0c  3597  uni0  3598
  Copyright terms: Public domain W3C validator