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Theorem ifpprsnss 4764
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 4734 . . . . . 6 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2 dfsn2 4637 . . . . . 6 {𝐴} = {𝐴, 𝐴}
31, 2eqtr4di 2785 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
43eqcoms 2735 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
54eqeq2d 2738 . . 3 (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴}))
65biimpac 478 . 2 ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴})
7 eqimss2 4037 . . 3 (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃)
87adantr 480 . 2 ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃)
96, 8ifpimpda 1079 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 1061   = wceq 1534  wss 3944  {csn 4624  {cpr 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-un 3949  df-in 3951  df-ss 3961  df-sn 4625  df-pr 4627
This theorem is referenced by:  upgriswlk  29442  eupth2lem3lem7  30031  upwlkwlk  47124
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