Theorem ifpprsnssdc 3774

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 3744	. . . . . . 7 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2		dfsn2 3680	. . . . . . 7 ⊢ {𝐴} = {𝐴, 𝐴}
3	1, 2	eqtr4di 2280	. . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴})
4	3	eqcoms 2232	. . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
5	4	eqeq2d 2241	. . . 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 3279	. . . 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 987	. . 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 950	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 839  if-wif 983   = wceq 1395   ⊆ wss 3197  {csn 3666  {cpr 3667

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 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673

This theorem is referenced by:  upgriswlkdc  16071