Theorem prssd 3763

Description: Deduction version of prssi 3762: 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 3762	. 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 2158   C_ wss 3141   {cpr 3605

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611

This theorem is referenced by:  0idnsgd  13116  lspprcl  13639  lsptpcl  13640  lspprss  13652  lspprid1  13657