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Theorem tpssg 24263
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
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 2316 . . . . 5  |-  ( a  =  A  ->  (
a  e.  D  <->  A  e.  D ) )
543anbi1d 1261 . . . 4  |-  ( a  =  A  ->  (
( a  e.  D  /\  b  e.  D  /\  c  e.  D
)  <->  ( A  e.  D  /\  b  e.  D  /\  c  e.  D ) ) )
6 tpeq1 3656 . . . . 5  |-  ( a  =  A  ->  { a ,  b ,  c }  =  { A ,  b ,  c } )
76sseq1d 3147 . . . 4  |-  ( a  =  A  ->  ( { a ,  b ,  c }  C_  D 
<->  { A ,  b ,  c }  C_  D ) )
85, 7bibi12d 314 . . 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 2316 . . . . 5  |-  ( b  =  B  ->  (
b  e.  D  <->  B  e.  D ) )
1093anbi2d 1262 . . . 4  |-  ( b  =  B  ->  (
( A  e.  D  /\  b  e.  D  /\  c  e.  D
)  <->  ( A  e.  D  /\  B  e.  D  /\  c  e.  D ) ) )
11 tpeq2 3657 . . . . 5  |-  ( b  =  B  ->  { A ,  b ,  c }  =  { A ,  B ,  c } )
1211sseq1d 3147 . . . 4  |-  ( b  =  B  ->  ( { A ,  b ,  c }  C_  D  <->  { A ,  B , 
c }  C_  D
) )
1310, 12bibi12d 314 . . 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 2316 . . . . 5  |-  ( c  =  C  ->  (
c  e.  D  <->  C  e.  D ) )
15143anbi3d 1263 . . . 4  |-  ( c  =  C  ->  (
( A  e.  D  /\  B  e.  D  /\  c  e.  D
)  <->  ( A  e.  D  /\  B  e.  D  /\  C  e.  D ) ) )
16 tpeq3 3658 . . . . 5  |-  ( c  =  C  ->  { A ,  B ,  c }  =  { A ,  B ,  C }
)
1716sseq1d 3147 . . . 4  |-  ( c  =  C  ->  ( { A ,  B , 
c }  C_  D  <->  { A ,  B ,  C }  C_  D ) )
1815, 17bibi12d 314 . . 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 2743 . . . . 5  |-  a  e. 
_V
20 vex 2743 . . . . 5  |-  b  e. 
_V
21 vex 2743 . . . . 5  |-  c  e. 
_V
2219, 20, 21tpss 3720 . . . 4  |-  ( ( a  e.  D  /\  b  e.  D  /\  c  e.  D )  <->  { a ,  b ,  c }  C_  D
)
2322a1i 12 . . 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 2804 . 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 1187 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 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621    C_ wss 3094   {ctp 3583
This theorem is referenced by:  isibg1a6  25457
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-un 3099  df-in 3101  df-ss 3108  df-sn 3587  df-pr 3588  df-tp 3589
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