Theorem dfopg 3802

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 2771	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 2771	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		df-3an 982	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
4	3	baibr 921	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
5	4	abbidv 2311	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}} = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})})
6		abid2 2314	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}} = {{𝐴}, {𝐴, 𝐵}}
7		df-op 3627	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
8	7	eqcomi 2197	. . . 4 ⊢ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ⟨𝐴, 𝐵⟩
9	5, 6, 8	3eqtr3g 2249	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} = ⟨𝐴, 𝐵⟩)
10	9	eqcomd 2199	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
11	1, 2, 10	syl2an 289	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  {cab 2179  Vcvv 2760  {csn 3618  {cpr 3619  ⟨cop 3621

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762  df-op 3627

This theorem is referenced by:  dfop  3803  opexg  4257  opth1  4265  opth  4266  0nelop  4277  op1stbg  4510