Theorem ssprss 3800

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 3796	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. { C , D } /\ B e. { C , D } ) <-> { A , B } C_ { C , D } ) )
2		elprg 3658	. . 3 |- ( A e. V -> ( A e. { C , D } <-> ( A = C \/ A = D ) ) )
3		elprg 3658	. . 3 |- ( B e. W -> ( B e. { C , D } <-> ( B = C \/ B = D ) ) )
4	2, 3	bi2anan9 606	. 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 710   = wceq 1373   e. wcel 2177   C_ wss 3170   {cpr 3639

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645

This theorem is referenced by:  ssprsseq  3801