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Theorem 0neqopab 6683
Description: The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
Assertion
Ref Expression
0neqopab ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem 0neqopab
StepHypRef Expression
1 elopab 4973 . . 3 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 nfopab1 4710 . . . . . 6 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
32nfel2 2778 . . . . 5 𝑥∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
43nfn 1782 . . . 4 𝑥 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5 nfopab2 4711 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
65nfel2 2778 . . . . . 6 𝑦∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
76nfn 1782 . . . . 5 𝑦 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
8 vex 3198 . . . . . . . 8 𝑥 ∈ V
9 vex 3198 . . . . . . . 8 𝑦 ∈ V
108, 9opnzi 4933 . . . . . . 7 𝑥, 𝑦⟩ ≠ ∅
11 nesym 2847 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ≠ ∅ ↔ ¬ ∅ = ⟨𝑥, 𝑦⟩)
12 pm2.21 120 . . . . . . . 8 (¬ ∅ = ⟨𝑥, 𝑦⟩ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
1311, 12sylbi 207 . . . . . . 7 (⟨𝑥, 𝑦⟩ ≠ ∅ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
1410, 13ax-mp 5 . . . . . 6 (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
1514adantr 481 . . . . 5 ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
167, 15exlimi 2084 . . . 4 (∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
174, 16exlimi 2084 . . 3 (∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
181, 17sylbi 207 . 2 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19 id 22 . 2 (¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2018, 19pm2.61i 176 1 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  wne 2791  c0 3907  cop 4174  {copab 4703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704
This theorem is referenced by:  brabv  6684  bj-0nelmpt  33044
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