Theorem ifpsnprss 16040

Description: Lemma for wlkvtxeledgg 16041: Two adjacent (not necessarily different) vertices A and B in a walk are incident with an edge E. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.)

Assertion

Ref	Expression
ifpsnprss	|- (if- ( A = B , E = { A } , { A , B } C_ E ) -> { A , B } C_ E )

Proof of Theorem ifpsnprss

Step	Hyp	Ref	Expression
1		ssidd 3245	. . 3 |- ( ( A = B /\ E = { A } ) -> { A } C_ { A } )
2		preq2 3744	. . . . . 6 |- ( B = A -> { A , B } = { A , A } )
3		dfsn2 3680	. . . . . 6 |- { A } = { A , A }
4	2, 3	eqtr4di 2280	. . . . 5 |- ( B = A -> { A , B } = { A } )
5	4	eqcoms 2232	. . . 4 |- ( A = B -> { A , B } = { A } )
6	5	adantr 276	. . 3 |- ( ( A = B /\ E = { A } ) -> { A , B } = { A } )
7		simpr 110	. . 3 |- ( ( A = B /\ E = { A } ) -> E = { A } )
8	1, 6, 7	3sstr4d 3269	. 2 |- ( ( A = B /\ E = { A } ) -> { A , B } C_ E )
9	8	1fpid3 1000	1 |- (if- ( A = B , E = { A } , { A , B } C_ E ) -> { A , B } C_ E )

Syntax hints:   -> wi 4   /\ wa 104  if-wif 983   = wceq 1395   C_ wss 3197   {csn 3666   {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-ifp 984  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:  wlkvtxeledgg  16041