Theorem dfop 4821

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 4820	. 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 2111  Vcvv 3436  {csn 4573  {cpr 4575  ⟨cop 4579

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 2113  ax-9 2121  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-ss 3914  df-nul 4281  df-if 4473  df-op 4580

This theorem is referenced by:  opi1  5406  opi2  5407  op1stb  5409  opeqpr  5443  propssopi  5446  uniop  5453  xpsspw  5748  relop  5789  funopg  6515