Theorem preqsn 4821

Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Revised by AV, 12-Jun-2022.)

Hypotheses

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

Assertion

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

Proof of Theorem preqsn

Step	Hyp	Ref	Expression
1		preqsn.1	. . 3 ⊢ 𝐴 ∈ V
2		id 23	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
3		preqsn.2	. . . . 5 ⊢ 𝐵 ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ V → 𝐵 ∈ V)
5	2, 4	preqsnd 4818	. . 3 ⊢ (𝐴 ∈ V → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))
6	1, 5	ax-mp 5	. 2 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
7		eqeq2 2772	. . 3 ⊢ (𝐵 = 𝐶 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶))
8	7	pm5.32ri 586	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
9	6, 8	bitr4i 281	1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {csn 4583  {cpr 4585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4584  df-pr 4586

This theorem is used by:  opeqsng  5474  propeqop  5478  propssopi  5479  relop  5827  hash2prde  14615  symg2bas  19607