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Theorem opabn0 4931
 Description: Nonempty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.)
Assertion
Ref Expression
opabn0 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝑦𝜑)

Proof of Theorem opabn0
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 n0 3890 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2 elopab 4908 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
32exbii 1764 . . 3 (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑧𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 exrot3 2032 . . . 4 (∃𝑧𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 opex 4859 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
65isseti 3182 . . . . . 6 𝑧 𝑧 = ⟨𝑥, 𝑦
7 19.41v 1901 . . . . . 6 (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
86, 7mpbiran 955 . . . . 5 (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑)
982exbii 1765 . . . 4 (∃𝑥𝑦𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝜑)
104, 9bitri 263 . . 3 (∃𝑧𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝜑)
113, 10bitri 263 . 2 (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦𝜑)
121, 11bitri 263 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝑦𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ⟨cop 4131  {copab 4642 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  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-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644 This theorem is referenced by:  csbopab  4932  dvdsrval  18468  thlle  19860  bcthlem5  22933  lgsquadlem3  24907
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