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Theorem ifpsnprss 26753
Description: Lemma for wlkvtxeledg 26754: 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 3773 . . . 4 {𝐴} ⊆ {𝐴}
21a1i 11 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴})
3 preq2 4405 . . . . . 6 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
4 dfsn2 4329 . . . . . 6 {𝐴} = {𝐴, 𝐴}
53, 4syl6eqr 2823 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
65eqcoms 2779 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
76adantr 466 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴, 𝐵} = {𝐴})
8 simpr 471 . . 3 ((𝐴 = 𝐵𝐸 = {𝐴}) → 𝐸 = {𝐴})
92, 7, 83sstr4d 3797 . 2 ((𝐴 = 𝐵𝐸 = {𝐴}) → {𝐴, 𝐵} ⊆ 𝐸)
1091fpid3 1066 1 (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  if-wif 1049   = wceq 1631  wss 3723  {csn 4316  {cpr 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3728  df-in 3730  df-ss 3737  df-sn 4317  df-pr 4319
This theorem is referenced by:  wlkvtxeledg  26754
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