Theorem ifpprsnss 4299

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
ifpprsnss	|- ( P = { A , B } -> if- ( A = B , P = { A } , { A , B } C_ P ) )

Proof of Theorem ifpprsnss

Step	Hyp	Ref	Expression
1		preq2 4269	. . . . . 6 |- ( B = A -> { A , B } = { A , A } )
2		dfsn2 4190	. . . . . 6 |- { A } = { A , A }
3	1, 2	syl6eqr 2674	. . . . 5 |- ( B = A -> { A , B } = { A } )
4	3	eqcoms 2630	. . . 4 |- ( A = B -> { A , B } = { A } )
5	4	eqeq2d 2632	. . 3 |- ( A = B -> ( P = { A , B } <-> P = { A } ) )
6	5	biimpac 503	. 2 |- ( ( P = { A , B } /\ A = B ) -> P = { A } )
7		eqimss2 3658	. . 3 |- ( P = { A , B } -> { A , B } C_ P )
8	7	adantr 481	. 2 |- ( ( P = { A , B } /\ -. A = B ) -> { A , B } C_ P )
9	6, 8	ifpimpda 1028	1 |- ( P = { A , B } -> if- ( A = B , P = { A } , { A , B } C_ P ) )

Syntax hints:   -. wn 3   -> wi 4  if-wif 1012   = wceq 1483   C_ wss 3574   {csn 4177   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180

This theorem is referenced by:  upgriswlk  26537  eupth2lem3lem7  27094  upwlkwlk  41720