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Theorem ifpprsnss 4530
 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 4500 . . . . . 6 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2 dfsn2 4410 . . . . . 6 {𝐴} = {𝐴, 𝐴}
31, 2syl6eqr 2831 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
43eqcoms 2785 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
54eqeq2d 2787 . . 3 (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴}))
65biimpac 472 . 2 ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴})
7 eqimss2 3876 . . 3 (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃)
87adantr 474 . 2 ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃)
96, 8ifpimpda 1062 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 1046   = wceq 1601   ⊆ wss 3791  {csn 4397  {cpr 4399 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ifp 1047  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-v 3399  df-un 3796  df-in 3798  df-ss 3805  df-sn 4398  df-pr 4400 This theorem is referenced by:  upgriswlk  26988  eupth2lem3lem7  27638  upwlkwlk  42755
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