Theorem dfopg 3823

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 2785	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 2785	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		df-3an 983	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
4	3	baibr 922	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})))
5	4	abbidv 2324	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}} = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})})
6		abid2 2327	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}} = {{𝐴}, {𝐴, 𝐵}}
7		df-op 3647	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
8	7	eqcomi 2210	. . . 4 ⊢ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ⟨𝐴, 𝐵⟩
9	5, 6, 8	3eqtr3g 2262	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} = ⟨𝐴, 𝐵⟩)
10	9	eqcomd 2212	. 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 981   = wceq 1373   ∈ wcel 2177  {cab 2192  Vcvv 2773  {csn 3638  {cpr 3639  ⟨cop 3641

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-v 2775  df-op 3647

This theorem is referenced by:  dfop  3824  opexg  4280  opth1  4288  opth  4289  0nelop  4300  op1stbg  4534