Theorem ifpsnprss 16193

Description: Lemma for wlkvtxeledgg 16194: 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 3248	. . 3 |- ( ( A = B /\ E = { A } ) -> { A } C_ { A } )
2		preq2 3749	. . . . . 6 |- ( B = A -> { A , B } = { A , A } )
3		dfsn2 3683	. . . . . 6 |- { A } = { A , A }
4	2, 3	eqtr4di 2282	. . . . 5 |- ( B = A -> { A , B } = { A } )
5	4	eqcoms 2234	. . . 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 3272	. 2 |- ( ( A = B /\ E = { A } ) -> { A , B } C_ E )
9	8	1fpid3 1002	1 |- (if- ( A = B , E = { A } , { A , B } C_ E ) -> { A , B } C_ E )

Syntax hints:   -> wi 4   /\ wa 104  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-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-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:  wlkvtxeledgg  16194