Theorem prssd 3837

Description: Deduction version of prssi 3836: 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

Step	Hyp	Ref	Expression
1		prssd.1	. 2 |- ( ph -> A e. C )
2		prssd.2	. 2 |- ( ph -> B e. C )
3		prssi 3836	. 2 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> { A , B } C_ C )

Syntax hints:   -> wi 4   e. wcel 2202   C_ wss 3201   {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680

This theorem is referenced by:  bassetsnn  13219  0idnsgd  13883  isnzr2  14279  lspprcl  14489  lsptpcl  14490  lspprss  14502  lspprid1  14507  perfectlem2  15814  upgr1edc  16062  uspgr1edc  16181  eupth2lemsfi  16419