Theorem xpsnopab 48031

Description: A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.)

Assertion

Ref	Expression
xpsnopab	⊢ ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)}

Distinct variable groups:   𝐶,𝑎,𝑏   𝑋,𝑎,𝑏

Proof of Theorem xpsnopab

Step	Hyp	Ref	Expression
1		df-xp 5671	. 2 ⊢ ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶)}
2		velsn 4622	. . . 4 ⊢ (𝑎 ∈ {𝑋} ↔ 𝑎 = 𝑋)
3	2	anbi1i 624	. . 3 ⊢ ((𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶) ↔ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶))
4	3	opabbii 5190	. 2 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)}
5	1, 4	eqtri 2757	1 ⊢ ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)}

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {csn 4606  {copab 5185   × cxp 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-sn 4607  df-opab 5186  df-xp 5671

This theorem is referenced by:  xpiun  48032