Theorem dfop 4803

Description: Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.) (Avoid depending on this detail.)

Hypotheses

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

Assertion

Ref	Expression
dfop	⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}

Proof of Theorem dfop

Step	Hyp	Ref	Expression
1		dfop.1	. 2 ⊢ 𝐴 ∈ V
2		dfop.2	. 2 ⊢ 𝐵 ∈ V
3		dfopg 4802	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4	1, 2, 3	mp2an 698	1 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}

Syntax hints:   = wceq 1547   ∈ wcel 2119  Vcvv 3431  {csn 4555  {cpr 4557  ⟨cop 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-ss 3900  df-nul 4262  df-if 4455  df-op 4562

This theorem is referenced by:  opi1  5408  opi2  5409  op1stb  5411  opeqpr  5446  propssopi  5449  uniop  5456  xpsspw  5752  relop  5792  funopg  6519