Theorem preqsn 3702

Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Hypotheses

Ref	Expression
preqsn.1	⊢ 𝐴 ∈ V
preqsn.2	⊢ 𝐵 ∈ V
preqsn.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
preqsn	⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))

Proof of Theorem preqsn

Step	Hyp	Ref	Expression
1		dfsn2 3541	. . 3 ⊢ {𝐶} = {𝐶, 𝐶}
2	1	eqeq2i 2150	. 2 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
3		preqsn.1	. . . 4 ⊢ 𝐴 ∈ V
4		preqsn.2	. . . 4 ⊢ 𝐵 ∈ V
5		preqsn.3	. . . 4 ⊢ 𝐶 ∈ V
6	3, 4, 5, 5	preq12b 3697	. . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))
7		oridm 746	. . . 4 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
8		eqtr3 2159	. . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
9		simpr 109	. . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
10	8, 9	jca 304	. . . . 5 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
11		eqtr 2157	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶)
12		simpr 109	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
13	11, 12	jca 304	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
14	10, 13	impbii 125	. . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
15	7, 14	bitri 183	. . 3 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
16	6, 15	bitri 183	. 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
17	2, 16	bitri 183	1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∨ wo 697   = wceq 1331   ∈ wcel 1480  Vcvv 2686  {csn 3527  {cpr 3528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534

This theorem is referenced by:  opeqsn  4174  relop  4689