Theorem dfop 4830

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 4829	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4	1, 2, 3	mp2an 693	1 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582  {cpr 4584  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-ss 3920  df-nul 4288  df-if 4482  df-op 4589

This theorem is referenced by:  opi1  5424  opi2  5425  op1stb  5427  opeqpr  5461  propssopi  5464  uniop  5471  xpsspw  5766  relop  5807  funopg  6534