Theorem dfopg 4835

Description: Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)

Assertion

Ref	Expression
dfopg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})

Proof of Theorem dfopg

Step	Hyp	Ref	Expression
1		elex 3468	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3468	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		dfopif 4834	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
4		iftrue 4494	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = {{𝐴}, {𝐴, 𝐵}})
5	3, 4	eqtrid 2776	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
6	1, 2, 5	syl2an 596	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ∅c0 4296  ifcif 4488  {csn 4589  {cpr 4591  ⟨cop 4595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-dif 3917  df-ss 3931  df-nul 4297  df-if 4489  df-op 4596

This theorem is referenced by:  dfop  4836  opidg  4856  elopg  5426  opnz  5433  opth1  5435  opth  5436  0nelop  5456  opeqsng  5463  opwf  9765  rankopb  9805  wunop  10675  tskop  10724  gruop  10758