Theorem ssprss 3828

Description: A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.)

Assertion

Ref	Expression
ssprss	|- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ { C , D } <-> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) )

Proof of Theorem ssprss

Step	Hyp	Ref	Expression
1		prssg 3824	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. { C , D } /\ B e. { C , D } ) <-> { A , B } C_ { C , D } ) )
2		elprg 3686	. . 3 |- ( A e. V -> ( A e. { C , D } <-> ( A = C \/ A = D ) ) )
3		elprg 3686	. . 3 |- ( B e. W -> ( B e. { C , D } <-> ( B = C \/ B = D ) ) )
4	2, 3	bi2anan9 608	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. { C , D } /\ B e. { C , D } ) <-> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) )
5	1, 4	bitr3d 190	1 |- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ { C , D } <-> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   C_ wss 3197   {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673

This theorem is referenced by:  ssprsseq  3829