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Theorem ssint 3894
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 3183 . 2  |-  ( A 
C_  |^| B  <->  A. y  e.  A  y  e.  |^| B )
2 vex 2804 . . . 4  |-  y  e. 
_V
32elint2 3885 . . 3  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
43ralbii 2580 . 2  |-  ( A. y  e.  A  y  e.  |^| B  <->  A. y  e.  A  A. x  e.  B  y  e.  x )
5 ralcom 2713 . . 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 3183 . . . 4  |-  ( A 
C_  x  <->  A. y  e.  A  y  e.  x )
76ralbii 2580 . . 3  |-  ( A. x  e.  B  A  C_  x  <->  A. x  e.  B  A. y  e.  A  y  e.  x )
85, 7bitr4i 243 . 2  |-  ( A. y  e.  A  A. x  e.  B  y  e.  x  <->  A. x  e.  B  A  C_  x )
91, 4, 83bitri 262 1  |-  ( A 
C_  |^| B  <->  A. x  e.  B  A  C_  x
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1696   A.wral 2556    C_ wss 3165   |^|cint 3878
This theorem is referenced by:  ssintab  3895  ssintub  3896  iinpw  4006  trint  4144  oneqmini  4459  fint  5436  sorpssint  6303  iscard2  7625  coftr  7915  isf32lem2  7996  inttsk  8412  isacs1i  13575  mrelatglb  14303  fbfinnfr  17552  fclscmp  17741  dfrtrcl2  24060  fneint  26380  topmeet  26416  igenval2  26794  ismrcd1  26876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-int 3879
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