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Theorem xpsnopab 41555
 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 5044 . 2 ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏𝐶)}
2 velsn 4141 . . . 4 (𝑎 ∈ {𝑋} ↔ 𝑎 = 𝑋)
32anbi1i 727 . . 3 ((𝑎 ∈ {𝑋} ∧ 𝑏𝐶) ↔ (𝑎 = 𝑋𝑏𝐶))
43opabbii 4649 . 2 {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏𝐶)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋𝑏𝐶)}
51, 4eqtri 2632 1 ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋𝑏𝐶)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  {copab 4642   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sn 4126  df-opab 4644  df-xp 5044 This theorem is referenced by:  xpiun  41556
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