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Theorem ifpsnprss 26574
Description: Lemma for wlkvtxeledg 26575: 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 ssid 3657 . . . 4 {𝐴} ⊆ {𝐴}
21a1i 11 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴})
3 preq2 4301 . . . . . 6 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
4 dfsn2 4223 . . . . . 6 {𝐴} = {𝐴, 𝐴}
53, 4syl6eqr 2703 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
65eqcoms 2659 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
76adantr 480 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴, 𝐵} = {𝐴})
8 simpr 476 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → 𝐸 = {𝐴})
92, 7, 83sstr4d 3681 . 2 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴, 𝐵} ⊆ 𝐸)
1091fpid3 1049 1 (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  if-wif 1032   = wceq 1523  wss 3607  {csn 4210  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-pr 4213
This theorem is referenced by:  wlkvtxeledg  26575
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