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Theorem opabn0 5110
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 4039 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2 elopab 5087 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
32exbii 1887 . . 3 (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑧𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 exrot3 2158 . . . 4 (∃𝑧𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 opex 5037 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
65isseti 3313 . . . . . 6 𝑧 𝑧 = ⟨𝑥, 𝑦
7 19.41v 1990 . . . . . 6 (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
86, 7mpbiran 991 . . . . 5 (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑)
982exbii 1888 . . . 4 (∃𝑥𝑦𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝜑)
104, 9bitri 264 . . 3 (∃𝑧𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝜑)
113, 10bitri 264 . 2 (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦𝜑)
121, 11bitri 264 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1596  wex 1817  wcel 2103  wne 2896  c0 4023  cop 4291  {copab 4820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-opab 4821
This theorem is referenced by:  opab0  5111  csbopab  5112  dvdsrval  18766  thlle  20164  bcthlem5  23246  lgsquadlem3  25227  br1cosscnvxrn  34466
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