Theorem ifpprsnssdc 3783

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
ifpprsnssdc	⊢ ((𝑃 = {𝐴, 𝐵} ∧ DECID 𝐴 = 𝐵) → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃))

Proof of Theorem ifpprsnssdc

Step	Hyp	Ref	Expression
1		preq2 3753	. . . . . . 7 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2		dfsn2 3687	. . . . . . 7 ⊢ {𝐴} = {𝐴, 𝐴}
3	1, 2	eqtr4di 2282	. . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
4	3	eqcoms 2234	. . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
5	4	eqeq2d 2243	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴}))
6	5	biimpcd 159	. . 3 ⊢ (𝑃 = {𝐴, 𝐵} → (𝐴 = 𝐵 → 𝑃 = {𝐴}))
7	6	adantr 276	. 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ DECID 𝐴 = 𝐵) → (𝐴 = 𝐵 → 𝑃 = {𝐴}))
8		eqimss2 3283	. . . 4 ⊢ (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃)
9	8	a1d 22	. . 3 ⊢ (𝑃 = {𝐴, 𝐵} → (¬ 𝐴 = 𝐵 → {𝐴, 𝐵} ⊆ 𝑃))
10	9	adantr 276	. 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ DECID 𝐴 = 𝐵) → (¬ 𝐴 = 𝐵 → {𝐴, 𝐵} ⊆ 𝑃))
11		dfifp2dc 990	. . 3 ⊢ (DECID 𝐴 = 𝐵 → (if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃) ↔ ((𝐴 = 𝐵 → 𝑃 = {𝐴}) ∧ (¬ 𝐴 = 𝐵 → {𝐴, 𝐵} ⊆ 𝑃))))
12	11	adantl 277	. 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ DECID 𝐴 = 𝐵) → (if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃) ↔ ((𝐴 = 𝐵 → 𝑃 = {𝐴}) ∧ (¬ 𝐴 = 𝐵 → {𝐴, 𝐵} ⊆ 𝑃))))
13	7, 10, 12	mpbir2and 953	1 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ DECID 𝐴 = 𝐵) → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 842  if-wif 986   = wceq 1398   ⊆ wss 3201  {csn 3673  {cpr 3674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680

This theorem is referenced by:  upgriswlkdc  16301  eupth2lem3lem7fi  16415