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

Theorem uneqin 3756
Description: Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
uneqin  |-  ( ( A  u.  B )  =  ( A  i^i  B )  <->  A  =  B
)

Proof of Theorem uneqin
StepHypRef Expression
1 eqimss 3551 . . . 4  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  ( A  u.  B )  C_  ( A  i^i  B ) )
2 unss 3674 . . . . 5  |-  ( ( A  C_  ( A  i^i  B )  /\  B  C_  ( A  i^i  B
) )  <->  ( A  u.  B )  C_  ( A  i^i  B ) )
3 ssin 3716 . . . . . . 7  |-  ( ( A  C_  A  /\  A  C_  B )  <->  A  C_  ( A  i^i  B ) )
4 sstr 3507 . . . . . . 7  |-  ( ( A  C_  A  /\  A  C_  B )  ->  A  C_  B )
53, 4sylbir 213 . . . . . 6  |-  ( A 
C_  ( A  i^i  B )  ->  A  C_  B
)
6 ssin 3716 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  <->  B  C_  ( A  i^i  B ) )
7 simpl 457 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  ->  B  C_  A )
86, 7sylbir 213 . . . . . 6  |-  ( B 
C_  ( A  i^i  B )  ->  B  C_  A
)
95, 8anim12i 566 . . . . 5  |-  ( ( A  C_  ( A  i^i  B )  /\  B  C_  ( A  i^i  B
) )  ->  ( A  C_  B  /\  B  C_  A ) )
102, 9sylbir 213 . . . 4  |-  ( ( A  u.  B ) 
C_  ( A  i^i  B )  ->  ( A  C_  B  /\  B  C_  A ) )
111, 10syl 16 . . 3  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  ( A  C_  B  /\  B  C_  A ) )
12 eqss 3514 . . 3  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
1311, 12sylibr 212 . 2  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  A  =  B )
14 unidm 3643 . . . 4  |-  ( A  u.  A )  =  A
15 inidm 3703 . . . 4  |-  ( A  i^i  A )  =  A
1614, 15eqtr4i 2489 . . 3  |-  ( A  u.  A )  =  ( A  i^i  A
)
17 uneq2 3648 . . 3  |-  ( A  =  B  ->  ( A  u.  A )  =  ( A  u.  B ) )
18 ineq2 3690 . . 3  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
1916, 17, 183eqtr3a 2522 . 2  |-  ( A  =  B  ->  ( A  u.  B )  =  ( A  i^i  B ) )
2013, 19impbii 188 1  |-  ( ( A  u.  B )  =  ( A  i^i  B )  <->  A  =  B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    u. cun 3469    i^i cin 3470    C_ wss 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-in 3478  df-ss 3485
This theorem is referenced by:  uniintsn  4326
  Copyright terms: Public domain W3C validator