ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  un00 Unicode version

Theorem un00 3291
Description: Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
un00  |-  ( ( A  =  (/)  /\  B  =  (/) )  <->  ( A  u.  B )  =  (/) )

Proof of Theorem un00
StepHypRef Expression
1 uneq12 3120 . . 3  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  u.  B )  =  ( (/)  u.  (/) ) )
2 un0 3279 . . 3  |-  ( (/)  u.  (/) )  =  (/)
31, 2syl6eq 2104 . 2  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  u.  B )  =  (/) )
4 ssun1 3134 . . . . 5  |-  A  C_  ( A  u.  B
)
5 sseq2 2995 . . . . 5  |-  ( ( A  u.  B )  =  (/)  ->  ( A 
C_  ( A  u.  B )  <->  A  C_  (/) ) )
64, 5mpbii 140 . . . 4  |-  ( ( A  u.  B )  =  (/)  ->  A  C_  (/) )
7 ss0b 3284 . . . 4  |-  ( A 
C_  (/)  <->  A  =  (/) )
86, 7sylib 131 . . 3  |-  ( ( A  u.  B )  =  (/)  ->  A  =  (/) )
9 ssun2 3135 . . . . 5  |-  B  C_  ( A  u.  B
)
10 sseq2 2995 . . . . 5  |-  ( ( A  u.  B )  =  (/)  ->  ( B 
C_  ( A  u.  B )  <->  B  C_  (/) ) )
119, 10mpbii 140 . . . 4  |-  ( ( A  u.  B )  =  (/)  ->  B  C_  (/) )
12 ss0b 3284 . . . 4  |-  ( B 
C_  (/)  <->  B  =  (/) )
1311, 12sylib 131 . . 3  |-  ( ( A  u.  B )  =  (/)  ->  B  =  (/) )
148, 13jca 294 . 2  |-  ( ( A  u.  B )  =  (/)  ->  ( A  =  (/)  /\  B  =  (/) ) )
153, 14impbii 121 1  |-  ( ( A  =  (/)  /\  B  =  (/) )  <->  ( A  u.  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259    u. cun 2943    C_ wss 2945   (/)c0 3252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253
This theorem is referenced by:  undisj1  3307  undisj2  3308  disjpr2  3462
  Copyright terms: Public domain W3C validator