Theorem dfop 3882

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.)

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 3881	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4	1, 2, 3	mp2an 426	1 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}

Syntax hints:   = wceq 1398   ∈ wcel 2203  Vcvv 2813  {csn 3689  {cpr 3690  ⟨cop 3692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-v 2815  df-op 3698

This theorem is referenced by:  opid  3901  elop  4347  opi1  4348  opi2  4349  opeqsn  4369  opeqpr  4370  uniop  4372  op1stb  4599  xpsspw  4862  relop  4905  funopg  5386  funopsn  5860