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Theorem ifpsnprss 29656
Description: Lemma for wlkvtxeledg 29657: 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
StepHypRef Expression
1 ssidd 4019 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴})
2 preq2 4739 . . . . . 6 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
3 dfsn2 4644 . . . . . 6 {𝐴} = {𝐴, 𝐴}
42, 3eqtr4di 2793 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
54eqcoms 2743 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
65adantr 480 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴, 𝐵} = {𝐴})
7 simpr 484 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → 𝐸 = {𝐴})
81, 6, 73sstr4d 4043 . 2 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴, 𝐵} ⊆ 𝐸)
981fpid3 1081 1 (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  if-wif 1062   = wceq 1537  wss 3963  {csn 4631  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634
This theorem is referenced by:  wlkvtxeledg  29657
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