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Theorem ssiinf 3697
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/_ C
Assertion
Ref Expression
ssiinf  C 
C_  |^|_  C  C_

Proof of Theorem ssiinf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . 5 
_V
2 eliin 3653 . . . . 5  _V  |^|_
31, 2ax-mp 7 . . . 4  |^|_
43ralbii 2324 . . 3  C  |^|_  C
5 ssiinf.1 . . . 4  F/_ C
6 nfcv 2175 . . . 4  F/_
75, 6ralcomf 2465 . . 3  C  C
84, 7bitri 173 . 2  C  |^|_  C
9 dfss3 2929 . 2  C 
C_  |^|_  C  |^|_
10 dfss3 2929 . . 3  C 
C_  C
1110ralbii 2324 . 2  C  C_  C
128, 9, 113bitr4i 201 1  C 
C_  |^|_  C  C_
Colors of variables: wff set class
Syntax hints:   wb 98   wcel 1390   F/_wnfc 2162  wral 2300   _Vcvv 2551    C_ wss 2911   |^|_ciin 3649
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-iin 3651
This theorem is referenced by:  ssiin  3698  dmiin  4523
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