Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xpsnopab Structured version   Visualization version   GIF version

Theorem xpsnopab 42090
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 5149 . 2 ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏𝐶)}
2 velsn 4226 . . . 4 (𝑎 ∈ {𝑋} ↔ 𝑎 = 𝑋)
32anbi1i 731 . . 3 ((𝑎 ∈ {𝑋} ∧ 𝑏𝐶) ↔ (𝑎 = 𝑋𝑏𝐶))
43opabbii 4750 . 2 {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑋} ∧ 𝑏𝐶)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋𝑏𝐶)}
51, 4eqtri 2673 1 ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋𝑏𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wcel 2030  {csn 4210  {copab 4745   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sn 4211  df-opab 4746  df-xp 5149
This theorem is referenced by:  xpiun  42091
  Copyright terms: Public domain W3C validator