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Theorem xpsnopab 46614
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 = 𝑋𝑏𝐶)}
Distinct variable groups:   𝐶,𝑎,𝑏   𝑋,𝑎,𝑏

Proof of Theorem xpsnopab
StepHypRef Expression
1 df-xp 5682 . 2 ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏𝐶)}
2 velsn 4644 . . . 4 (𝑎 ∈ {𝑋} ↔ 𝑎 = 𝑋)
32anbi1i 624 . . 3 ((𝑎 ∈ {𝑋} ∧ 𝑏𝐶) ↔ (𝑎 = 𝑋𝑏𝐶))
43opabbii 5215 . 2 {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏𝐶)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋𝑏𝐶)}
51, 4eqtri 2760 1 ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋𝑏𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  {csn 4628  {copab 5210   × cxp 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-sn 4629  df-opab 5211  df-xp 5682
This theorem is referenced by:  xpiun  46615
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