Theorem ssprss 3839

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 3835	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. { C , D } /\ B e. { C , D } ) <-> { A , B } C_ { C , D } ) )
2		elprg 3693	. . 3 |- ( A e. V -> ( A e. { C , D } <-> ( A = C \/ A = D ) ) )
3		elprg 3693	. . 3 |- ( B e. W -> ( B e. { C , D } <-> ( B = C \/ B = D ) ) )
4	2, 3	bi2anan9 610	. 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 716   = wceq 1398   e. wcel 2202   C_ wss 3201   {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680

This theorem is referenced by:  ssprsseq  3840