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

Theorem ssiinf 3706
Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
ssiinf.1  |-  F/_ x C
Assertion
Ref Expression
ssiinf  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )

Proof of Theorem ssiinf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . . 5  |-  y  e. 
_V
2 eliin 3662 . . . . 5  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
31, 2ax-mp 7 . . . 4  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
43ralbii 2330 . . 3  |-  ( A. y  e.  C  y  e.  |^|_ x  e.  A  B 
<-> 
A. y  e.  C  A. x  e.  A  y  e.  B )
5 ssiinf.1 . . . 4  |-  F/_ x C
6 nfcv 2178 . . . 4  |-  F/_ y A
75, 6ralcomf 2471 . . 3  |-  ( A. y  e.  C  A. x  e.  A  y  e.  B  <->  A. x  e.  A  A. y  e.  C  y  e.  B )
84, 7bitri 173 . 2  |-  ( A. y  e.  C  y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  A. y  e.  C  y  e.  B )
9 dfss3 2935 . 2  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. y  e.  C  y  e.  |^|_ x  e.  A  B )
10 dfss3 2935 . . 3  |-  ( C 
C_  B  <->  A. y  e.  C  y  e.  B )
1110ralbii 2330 . 2  |-  ( A. x  e.  A  C  C_  B  <->  A. x  e.  A  A. y  e.  C  y  e.  B )
128, 9, 113bitr4i 201 1  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 98    e. wcel 1393   F/_wnfc 2165   A.wral 2306   _Vcvv 2557    C_ wss 2917   |^|_ciin 3658
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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-iin 3660
This theorem is referenced by:  ssiin  3707  dmiin  4580
  Copyright terms: Public domain W3C validator