Theorem ifpprsnss 4735

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

Step	Hyp	Ref	Expression
1		preq2 4705	. . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2		dfsn2 4607	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
3	1, 2	eqtr4di 2819	. . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
4	3	eqcoms 2774	. . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
5	4	eqeq2d 2777	. . 3 ⊢ (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴}))
6	5	biimpac 484	. 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴})
7		eqimss2 3999	. . 3 ⊢ (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃)
8	7	adantr 486	. 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃)
9	6, 8	ifpimpda 1097	1 ⊢ (𝑃 = {𝐴, 𝐵} → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  if-wif 1078   = wceq 1570   ⊆ wss 3908  {csn 4594  {cpr 4596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-un 3913  df-ss 3925  df-sn 4595  df-pr 4597

This theorem is used by:  upgriswlk  30027  eupth2lem3lem7  30622  upwlkwlk  48945