Theorem preqsn 4862

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 22	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
3		preqsn.2	. . . . 5 ⊢ 𝐵 ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ V → 𝐵 ∈ V)
5	2, 4	preqsnd 4859	. . 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 2745	. . 3 ⊢ (𝐵 = 𝐶 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶))
8	7	pm5.32ri 577	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
9	6, 8	bitr4i 278	1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  {csn 4628  {cpr 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-pr 4631

This theorem is referenced by:  opeqsng  5503  propeqop  5507  propssopi  5508  relop  5849  hash2prde  14428  symg2bas  19255