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Theorem ssint 4058
Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
ssint  |-  ( A 
C_  |^| B  <->  A. x  e.  B  A  C_  x
)
Distinct variable groups:    x, A    x, B

Proof of Theorem ssint
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfss3 3330 . 2  |-  ( A 
C_  |^| B  <->  A. y  e.  A  y  e.  |^| B )
2 vex 2951 . . . 4  |-  y  e. 
_V
32elint2 4049 . . 3  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
43ralbii 2721 . 2  |-  ( A. y  e.  A  y  e.  |^| B  <->  A. y  e.  A  A. x  e.  B  y  e.  x )
5 ralcom 2860 . . 3  |-  ( A. y  e.  A  A. x  e.  B  y  e.  x  <->  A. x  e.  B  A. y  e.  A  y  e.  x )
6 dfss3 3330 . . . 4  |-  ( A 
C_  x  <->  A. y  e.  A  y  e.  x )
76ralbii 2721 . . 3  |-  ( A. x  e.  B  A  C_  x  <->  A. x  e.  B  A. y  e.  A  y  e.  x )
85, 7bitr4i 244 . 2  |-  ( A. y  e.  A  A. x  e.  B  y  e.  x  <->  A. x  e.  B  A  C_  x )
91, 4, 83bitri 263 1  |-  ( A 
C_  |^| B  <->  A. x  e.  B  A  C_  x
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1725   A.wral 2697    C_ wss 3312   |^|cint 4042
This theorem is referenced by:  ssintab  4059  ssintub  4060  iinpw  4171  trint  4309  oneqmini  4624  fint  5613  sorpssint  6523  iscard2  7852  coftr  8142  isf32lem2  8223  inttsk  8638  isacs1i  13870  mrelatglb  14598  fbfinnfr  17861  fclscmp  18050  dfrtrcl2  25136  fneint  26294  topmeet  26330  igenval2  26613  ismrcd1  26689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-in 3319  df-ss 3326  df-int 4043
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