Theorem ifpsnprss 27970

Description: Lemma for wlkvtxeledg 27971: 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} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)

Proof of Theorem ifpsnprss

Step	Hyp	Ref	Expression
1		ssidd 3948	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴})
2		preq2 4675	. . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
3		dfsn2 4579	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
4	2, 3	eqtr4di 2797	. . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
5	4	eqcoms 2747	. . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
6	5	adantr 480	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} = {𝐴})
7		simpr 484	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → 𝐸 = {𝐴})
8	1, 6, 7	3sstr4d 3972	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} ⊆ 𝐸)
9	8	1fpid3 1080	1 ⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)

Syntax hints:   → wi 4   ∧ wa 395  if-wif 1059   = wceq 1541   ⊆ wss 3891  {csn 4566  {cpr 4568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-v 3432  df-un 3896  df-in 3898  df-ss 3908  df-sn 4567  df-pr 4569

This theorem is referenced by:  wlkvtxeledg  27971