Theorem ifpprsnss 4716

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 4686	. . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2		dfsn2 4588	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
3	1, 2	eqtr4di 2786	. . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
4	3	eqcoms 2741	. . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
5	4	eqeq2d 2744	. . 3 ⊢ (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴}))
6	5	biimpac 478	. 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴})
7		eqimss2 3990	. . 3 ⊢ (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃)
8	7	adantr 480	. 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃)
9	6, 8	ifpimpda 1080	1 ⊢ (𝑃 = {𝐴, 𝐵} → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃))

Syntax hints:  ¬ wn 3   → wi 4  if-wif 1062   = wceq 1541   ⊆ wss 3898  {csn 4575  {cpr 4577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-un 3903  df-ss 3915  df-sn 4576  df-pr 4578

This theorem is referenced by:  upgriswlk  29621  eupth2lem3lem7  30216  upwlkwlk  48263