Theorem dfopg 3756

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

Assertion

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

Proof of Theorem dfopg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2737	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 2737	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		df-3an 970	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
4	3	baibr 910	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
5	4	abbidv 2284	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}} = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})})
6		abid2 2287	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}} = {{𝐴}, {𝐴, 𝐵}}
7		df-op 3585	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
8	7	eqcomi 2169	. . . 4 ⊢ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ⟨𝐴, 𝐵⟩
9	5, 6, 8	3eqtr3g 2222	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} = ⟨𝐴, 𝐵⟩)
10	9	eqcomd 2171	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
11	1, 2, 10	syl2an 287	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  {cab 2151  Vcvv 2726  {csn 3576  {cpr 3577  ⟨cop 3579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728  df-op 3585

This theorem is referenced by:  dfop  3757  opexg  4206  opth1  4214  opth  4215  0nelop  4226  op1stbg  4457