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Theorem ifpprsnss 4271
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 4241 . . . . . 6 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2 dfsn2 4163 . . . . . 6 {𝐴} = {𝐴, 𝐴}
31, 2syl6eqr 2673 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
43eqcoms 2629 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
54eqeq2d 2631 . . 3 (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴}))
65biimpac 503 . 2 ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴})
7 eqimss2 3639 . . 3 (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃)
87adantr 481 . 2 ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃)
96, 8ifpimpda 1027 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 1011   = wceq 1480  wss 3556  {csn 4150  {cpr 4152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3561  df-in 3563  df-ss 3570  df-sn 4151  df-pr 4153
This theorem is referenced by:  upgriswlk  26413  eupth2lem3lem7  26967  upwlkwlk  41024
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