Theorem xpsnopab 48905

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 5669	. 2 ⊢ ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶)}
2		velsn 4606	. . . 4 ⊢ (𝑎 ∈ {𝑋} ↔ 𝑎 = 𝑋)
3	2	anbi1i 635	. . 3 ⊢ ((𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶) ↔ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶))
4	3	opabbii 5179	. 2 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)}
5	1, 4	eqtri 2786	1 ⊢ ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)}

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {csn 4590  {copab 5174   × cxp 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-sn 4591  df-opab 5175  df-xp 5669

This theorem is referenced by:  xpiun  48906