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Theorem intmin4 3634
Description: Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
Assertion
Ref Expression
intmin4 
C_  |^| {  |  }  |^| {  |  C_  }  |^|
{  |  }
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem intmin4
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssintab 3623 . . . 4 
C_  |^| {  |  }  C_
2 simpr 103 . . . . . . . 8  C_
3 ancr 304 . . . . . . . 8  C_  C_
42, 3impbid2 131 . . . . . . 7  C_ 
C_
54imbi1d 220 . . . . . 6  C_  C_
65alimi 1341 . . . . 5  C_ 
C_
7 albi 1354 . . . . 5 
C_  C_
86, 7syl 14 . . . 4  C_ 
C_
91, 8sylbi 114 . . 3 
C_  |^| {  |  }  C_
10 vex 2554 . . . 4 
_V
1110elintab 3617 . . 3  |^| {  |  C_  }  C_
1210elintab 3617 . . 3  |^| {  |  }
139, 11, 123bitr4g 212 . 2 
C_  |^| {  |  }  |^| {  |  C_  } 
|^| {  |  }
1413eqrdv 2035 1 
C_  |^| {  |  }  |^| {  |  C_  }  |^|
{  |  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242   wcel 1390   {cab 2023    C_ wss 2911   |^|cint 3606
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-int 3607
This theorem is referenced by: (None)
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