Theorem ifpsnprss 26518

Description: Lemma for wlkvtxeledg 26519: 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		ssid 3624	. . . 4 |- { A } C_ { A }
2	1	a1i 11	. . 3 |- ( ( A = B /\ E = { A } ) -> { A } C_ { A } )
3		preq2 4269	. . . . . 6 |- ( B = A -> { A , B } = { A , A } )
4		dfsn2 4190	. . . . . 6 |- { A } = { A , A }
5	3, 4	syl6eqr 2674	. . . . 5 |- ( B = A -> { A , B } = { A } )
6	5	eqcoms 2630	. . . 4 |- ( A = B -> { A , B } = { A } )
7	6	adantr 481	. . 3 |- ( ( A = B /\ E = { A } ) -> { A , B } = { A } )
8		simpr 477	. . 3 |- ( ( A = B /\ E = { A } ) -> E = { A } )
9	2, 7, 8	3sstr4d 3648	. 2 |- ( ( A = B /\ E = { A } ) -> { A , B } C_ E )
10	9	1fpid3 1029	1 |- (if- ( A = B , E = { A } , { A , B } C_ E ) -> { A , B } C_ E )

Syntax hints:   -> wi 4   /\ wa 384  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:  wlkvtxeledg  26519