Theorem ifpprsnssdc 3779

Description: An unordered pair is a singleton or a subset of itself. This theorem is helpful to convert theorems about walks in arbitrary graphs into theorems about walks in pseudographs. (Contributed by AV, 27-Feb-2021.)

Assertion

Ref	Expression
ifpprsnssdc	|- ( ( P = { A , B } /\ DECID A = B ) -> if- ( A = B , P = { A } , { A , B } C_ P ) )

Proof of Theorem ifpprsnssdc

Step	Hyp	Ref	Expression
1		preq2 3749	. . . . . . 7 |- ( B = A -> { A , B } = { A , A } )
2		dfsn2 3683	. . . . . . 7 |- { A } = { A , A }
3	1, 2	eqtr4di 2282	. . . . . 6 |- ( B = A -> { A , B } = { A } )
4	3	eqcoms 2234	. . . . 5 |- ( A = B -> { A , B } = { A } )
5	4	eqeq2d 2243	. . . 4 |- ( A = B -> ( P = { A , B } <-> P = { A } ) )
6	5	biimpcd 159	. . 3 |- ( P = { A , B } -> ( A = B -> P = { A } ) )
7	6	adantr 276	. 2 |- ( ( P = { A , B } /\ DECID A = B ) -> ( A = B -> P = { A } ) )
8		eqimss2 3282	. . . 4 |- ( P = { A , B } -> { A , B } C_ P )
9	8	a1d 22	. . 3 |- ( P = { A , B } -> ( -. A = B -> { A , B } C_ P ) )
10	9	adantr 276	. 2 |- ( ( P = { A , B } /\ DECID A = B ) -> ( -. A = B -> { A , B } C_ P ) )
11		dfifp2dc 989	. . 3 |- (DECID A = B -> (if- ( A = B , P = { A } , { A , B } C_ P ) <-> ( ( A = B -> P = { A } ) /\ ( -. A = B -> { A , B } C_ P ) ) ) )
12	11	adantl 277	. 2 |- ( ( P = { A , B } /\ DECID A = B ) -> (if- ( A = B , P = { A } , { A , B } C_ P ) <-> ( ( A = B -> P = { A } ) /\ ( -. A = B -> { A , B } C_ P ) ) ) )
13	7, 10, 12	mpbir2and 952	1 |- ( ( P = { A , B } /\ DECID A = B ) -> if- ( A = B , P = { A } , { A , B } C_ P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 841  if-wif 985   = wceq 1397   C_ wss 3200   {csn 3669   {cpr 3670

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-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-ifp 986  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676

This theorem is referenced by:  upgriswlkdc  16210