Theorem ifpsnprss 29605

Description: Lemma for wlkvtxeledg 29606: 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 3954	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴})
2		preq2 4688	. . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
3		dfsn2 4590	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
4	2, 3	eqtr4di 2786	. . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
5	4	eqcoms 2741	. . . 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 3986	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} ⊆ 𝐸)
9	8	1fpid3 1081	1 ⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)

Syntax hints:   → wi 4   ∧ wa 395  if-wif 1062   = wceq 1541   ⊆ wss 3898  {csn 4577  {cpr 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-un 3903  df-ss 3915  df-sn 4578  df-pr 4580

This theorem is referenced by:  wlkvtxeledg  29606