Theorem dfop 4826

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 4825	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4	1, 2, 3	mp2an 692	1 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3438  {csn 4578  {cpr 4580  ⟨cop 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-dif 3902  df-ss 3916  df-nul 4284  df-if 4478  df-op 4585

This theorem is referenced by:  opi1  5414  opi2  5415  op1stb  5417  opeqpr  5451  propssopi  5454  uniop  5461  xpsspw  5756  relop  5797  funopg  6524