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Theorem ifpsnprss 28279
Description: Lemma for wlkvtxeledg 28280: 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 3955 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴})
2 preq2 4682 . . . . . 6 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
3 dfsn2 4586 . . . . . 6 {𝐴} = {𝐴, 𝐴}
42, 3eqtr4di 2794 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
54eqcoms 2744 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
65adantr 481 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴, 𝐵} = {𝐴})
7 simpr 485 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → 𝐸 = {𝐴})
81, 6, 73sstr4d 3979 . 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 396  if-wif 1060   = wceq 1540  wss 3898  {csn 4573  {cpr 4575
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-un 3903  df-in 3905  df-ss 3915  df-sn 4574  df-pr 4576
This theorem is referenced by:  wlkvtxeledg  28280
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