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Theorem ifpprsnss 4436
Description: An unordered pair is a singleton or a subset of itself. This theorem is helpful to convert theorems about walks in arbitrary graphs into theorems about walks in pseudographs. (Contributed by AV, 27-Feb-2021.)
Assertion
Ref Expression
ifpprsnss (𝑃 = {𝐴, 𝐵} → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃))

Proof of Theorem ifpprsnss
StepHypRef Expression
1 preq2 4406 . . . . . 6 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2 dfsn2 4330 . . . . . 6 {𝐴} = {𝐴, 𝐴}
31, 2syl6eqr 2823 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
43eqcoms 2779 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
54eqeq2d 2781 . . 3 (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴}))
65biimpac 464 . 2 ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴})
7 eqimss2 3808 . . 3 (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃)
87adantr 466 . 2 ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃)
96, 8ifpimpda 1065 1 (𝑃 = {𝐴, 𝐵} → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  if-wif 1049   = wceq 1631  wss 3724  {csn 4317  {cpr 4319
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 829  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 3729  df-in 3731  df-ss 3738  df-sn 4318  df-pr 4320
This theorem is referenced by:  upgriswlk  26773  eupth2lem3lem7  27415  upwlkwlk  42249
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