Theorem prssd 4354

Description: Deduction version of prssi 4353: 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 4353	. 2 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
4	1, 2, 3	syl2anc 693	1 |- ( ph -> { A , B } C_ C )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180

This theorem is referenced by:  nehash2  13256  basprssdmsets  15925  dchrisum0re  25202  upgrex  25987  upgr1e  26008  uspgr1e  26136  eupth2lems  27098  pmtridf1o  29856  poimirlem9  33418  clsk1indlem4  38342  clsk1indlem1  38343  limsup10exlem  40004  meadjun  40679