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Theorem prssd 3777
Description: Deduction version of prssi 3776: A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
prssd.1  |-  ( ph  ->  A  e.  C )
prssd.2  |-  ( ph  ->  B  e.  C )
Assertion
Ref Expression
prssd  |-  ( ph  ->  { A ,  B }  C_  C )

Proof of Theorem prssd
StepHypRef Expression
1 prssd.1 . 2  |-  ( ph  ->  A  e.  C )
2 prssd.2 . 2  |-  ( ph  ->  B  e.  C )
3 prssi 3776 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
41, 2, 3syl2anc 411 1  |-  ( ph  ->  { A ,  B }  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2164    C_ wss 3153   {cpr 3619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625
This theorem is referenced by:  0idnsgd  13275  isnzr2  13664  lspprcl  13873  lsptpcl  13874  lspprss  13886  lspprid1  13891
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