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Theorem ifpsnprss 28880
Description: Lemma for wlkvtxeledg 28881: 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 4006 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴})
2 preq2 4739 . . . . . 6 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
3 dfsn2 4642 . . . . . 6 {𝐴} = {𝐴, 𝐴}
42, 3eqtr4di 2791 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
54eqcoms 2741 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
65adantr 482 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴, 𝐵} = {𝐴})
7 simpr 486 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → 𝐸 = {𝐴})
81, 6, 73sstr4d 4030 . 2 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴, 𝐵} ⊆ 𝐸)
981fpid3 1083 1 (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  if-wif 1062   = wceq 1542  wss 3949  {csn 4629  {cpr 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-pr 4632
This theorem is referenced by:  wlkvtxeledg  28881
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