Theorem xpsnopab 48648

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 5624	. 2 ⊢ ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶)}
2		velsn 4571	. . . 4 ⊢ (𝑎 ∈ {𝑋} ↔ 𝑎 = 𝑋)
3	2	anbi1i 630	. . 3 ⊢ ((𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶) ↔ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶))
4	3	opabbii 5139	. 2 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)}
5	1, 4	eqtri 2762	1 ⊢ ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)}

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {csn 4555  {copab 5134   × cxp 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-sn 4556  df-opab 5135  df-xp 5624

This theorem is referenced by:  xpiun  48649