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Theorem ifpprsnss 4769
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 4739 . . . . . 6 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2 dfsn2 4644 . . . . . 6 {𝐴} = {𝐴, 𝐴}
31, 2eqtr4di 2793 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
43eqcoms 2743 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
54eqeq2d 2746 . . 3 (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴}))
65biimpac 478 . 2 ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴})
7 eqimss2 4055 . . 3 (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃)
87adantr 480 . 2 ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃)
96, 8ifpimpda 1080 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 1062   = wceq 1537  wss 3963  {csn 4631  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634
This theorem is referenced by:  upgriswlk  29674  eupth2lem3lem7  30263  upwlkwlk  47983
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