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Theorem 0neqopab 6864
 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 5133 . . 3 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 nfopab1 4871 . . . . . 6 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
32nfel2 2919 . . . . 5 𝑥∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
43nfn 1933 . . . 4 𝑥 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5 nfopab2 4872 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
65nfel2 2919 . . . . . 6 𝑦∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
76nfn 1933 . . . . 5 𝑦 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
8 vex 3343 . . . . . . . 8 𝑥 ∈ V
9 vex 3343 . . . . . . . 8 𝑦 ∈ V
108, 9opnzi 5091 . . . . . . 7 𝑥, 𝑦⟩ ≠ ∅
11 nesym 2988 . . . . . . . 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 472 . . . . 5 ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
167, 15exlimi 2233 . . . 4 (∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
174, 16exlimi 2233 . . 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 383   = wceq 1632  ∃wex 1853   ∈ wcel 2139   ≠ wne 2932  ∅c0 4058  ⟨cop 4327  {copab 4864 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865 This theorem is referenced by:  brabv  6865  bj-0nelmpt  33393
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