Theorem ifpsnprss 27402

Description: Lemma for wlkvtxeledg 27403: 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 3983	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴})
2		preq2 4663	. . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
3		dfsn2 4573	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
4	2, 3	syl6eqr 2873	. . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
5	4	eqcoms 2828	. . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
6	5	adantr 483	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} = {𝐴})
7		simpr 487	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → 𝐸 = {𝐴})
8	1, 6, 7	3sstr4d 4007	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} ⊆ 𝐸)
9	8	1fpid3 1077	1 ⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)

Syntax hints:   → wi 4   ∧ wa 398  if-wif 1057   = wceq 1536   ⊆ wss 3929  {csn 4560  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-v 3493  df-un 3934  df-in 3936  df-ss 3945  df-sn 4561  df-pr 4563

This theorem is referenced by:  wlkvtxeledg  27403