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Theorem tpssg 24943
Description: A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Revised by FL, 17-May-2016.)
Hypotheses
Ref Expression
tpssg.1  |-  ( ph  ->  A  e.  E )
tpssg.2  |-  ( ph  ->  B  e.  F )
tpssg.3  |-  ( ph  ->  C  e.  G )
Assertion
Ref Expression
tpssg  |-  ( ph  ->  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D ) )

Proof of Theorem tpssg
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tpssg.1 . 2  |-  ( ph  ->  A  e.  E )
2 tpssg.2 . 2  |-  ( ph  ->  B  e.  F )
3 tpssg.3 . 2  |-  ( ph  ->  C  e.  G )
4 eleq1 2345 . . . . 5  |-  ( a  =  A  ->  (
a  e.  D  <->  A  e.  D ) )
543anbi1d 1256 . . . 4  |-  ( a  =  A  ->  (
( a  e.  D  /\  b  e.  D  /\  c  e.  D
)  <->  ( A  e.  D  /\  b  e.  D  /\  c  e.  D ) ) )
6 tpeq1 3717 . . . . 5  |-  ( a  =  A  ->  { a ,  b ,  c }  =  { A ,  b ,  c } )
76sseq1d 3207 . . . 4  |-  ( a  =  A  ->  ( { a ,  b ,  c }  C_  D 
<->  { A ,  b ,  c }  C_  D ) )
85, 7bibi12d 312 . . 3  |-  ( a  =  A  ->  (
( ( a  e.  D  /\  b  e.  D  /\  c  e.  D )  <->  { a ,  b ,  c }  C_  D )  <->  ( ( A  e.  D  /\  b  e.  D  /\  c  e.  D
)  <->  { A ,  b ,  c }  C_  D ) ) )
9 eleq1 2345 . . . . 5  |-  ( b  =  B  ->  (
b  e.  D  <->  B  e.  D ) )
1093anbi2d 1257 . . . 4  |-  ( b  =  B  ->  (
( A  e.  D  /\  b  e.  D  /\  c  e.  D
)  <->  ( A  e.  D  /\  B  e.  D  /\  c  e.  D ) ) )
11 tpeq2 3718 . . . . 5  |-  ( b  =  B  ->  { A ,  b ,  c }  =  { A ,  B ,  c } )
1211sseq1d 3207 . . . 4  |-  ( b  =  B  ->  ( { A ,  b ,  c }  C_  D  <->  { A ,  B , 
c }  C_  D
) )
1310, 12bibi12d 312 . . 3  |-  ( b  =  B  ->  (
( ( A  e.  D  /\  b  e.  D  /\  c  e.  D )  <->  { A ,  b ,  c }  C_  D )  <->  ( ( A  e.  D  /\  B  e.  D  /\  c  e.  D
)  <->  { A ,  B ,  c }  C_  D ) ) )
14 eleq1 2345 . . . . 5  |-  ( c  =  C  ->  (
c  e.  D  <->  C  e.  D ) )
15143anbi3d 1258 . . . 4  |-  ( c  =  C  ->  (
( A  e.  D  /\  B  e.  D  /\  c  e.  D
)  <->  ( A  e.  D  /\  B  e.  D  /\  C  e.  D ) ) )
16 tpeq3 3719 . . . . 5  |-  ( c  =  C  ->  { A ,  B ,  c }  =  { A ,  B ,  C }
)
1716sseq1d 3207 . . . 4  |-  ( c  =  C  ->  ( { A ,  B , 
c }  C_  D  <->  { A ,  B ,  C }  C_  D ) )
1815, 17bibi12d 312 . . 3  |-  ( c  =  C  ->  (
( ( A  e.  D  /\  B  e.  D  /\  c  e.  D )  <->  { A ,  B ,  c } 
C_  D )  <->  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D ) ) )
19 vex 2793 . . . . 5  |-  a  e. 
_V
20 vex 2793 . . . . 5  |-  b  e. 
_V
21 vex 2793 . . . . 5  |-  c  e. 
_V
2219, 20, 21tpss 3781 . . . 4  |-  ( ( a  e.  D  /\  b  e.  D  /\  c  e.  D )  <->  { a ,  b ,  c }  C_  D
)
2322a1i 10 . . 3  |-  ( ( a  e.  E  /\  b  e.  F  /\  c  e.  G )  ->  ( ( a  e.  D  /\  b  e.  D  /\  c  e.  D )  <->  { a ,  b ,  c }  C_  D )
)
248, 13, 18, 23vtocl3ga 2855 . 2  |-  ( ( A  e.  E  /\  B  e.  F  /\  C  e.  G )  ->  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D ) )
251, 2, 3, 24syl3anc 1182 1  |-  ( ph  ->  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1625    e. wcel 1686    C_ wss 3154   {ctp 3644
This theorem is referenced by:  isibg1a6  26136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-un 3159  df-in 3161  df-ss 3168  df-sn 3648  df-pr 3649  df-tp 3650
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