Theorem prssd 3874

Description: Deduction version of prssi 3873: 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 3873	. 2 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
4	1, 2, 3	syl2anc 415	1 |- ( ph -> { A , B } C_ C )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   C_ wss 3220   {cpr 3710

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716

This theorem is used by:  bassetsnn  13409  0idnsgd  14019  isnzr2  14491  lspprcl  14730  lsptpcl  14731  lspprss  14743  lspprid1  14748  perfectlem2  16114  upgr1edc  16362  uspgr1edc  16481  eupth2lemsfi  16719