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Theorem prss 3548
Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypotheses
Ref Expression
prss.1  |-  A  e. 
_V
prss.2  |-  B  e. 
_V
Assertion
Ref Expression
prss  |-  ( ( A  e.  C  /\  B  e.  C )  <->  { A ,  B }  C_  C )

Proof of Theorem prss
StepHypRef Expression
1 unss 3145 . 2  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  ( { A }  u.  { B } )  C_  C
)
2 prss.1 . . . 4  |-  A  e. 
_V
32snss 3522 . . 3  |-  ( A  e.  C  <->  { A }  C_  C )
4 prss.2 . . . 4  |-  B  e. 
_V
54snss 3522 . . 3  |-  ( B  e.  C  <->  { B }  C_  C )
63, 5anbi12i 441 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  <->  ( { A }  C_  C  /\  { B }  C_  C ) )
7 df-pr 3410 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
87sseq1i 2997 . 2  |-  ( { A ,  B }  C_  C  <->  ( { A }  u.  { B } )  C_  C
)
91, 6, 83bitr4i 205 1  |-  ( ( A  e.  C  /\  B  e.  C )  <->  { A ,  B }  C_  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    e. wcel 1409   _Vcvv 2574    u. cun 2943    C_ wss 2945   {csn 3403   {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410
This theorem is referenced by:  tpss  3557  prsspw  3564
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