Theorem preqsn 4789

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 4786	. . 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 2750	. . 3 ⊢ (𝐵 = 𝐶 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶))
8	7	pm5.32ri 575	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
9	6, 8	bitr4i 277	1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-sn 4559  df-pr 4561

This theorem is referenced by:  opeqsng  5411  propeqop  5415  propssopi  5416  relop  5748  hash2prde  14112  symg2bas  18915