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Theorem 0neqopab 6574
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 4898 . . 3 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 nfopab1 4645 . . . . . 6 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
32nfel2 2766 . . . . 5 𝑥∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
43nfn 1767 . . . 4 𝑥 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5 nfopab2 4646 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
65nfel2 2766 . . . . . 6 𝑦∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
76nfn 1767 . . . . 5 𝑦 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
8 vex 3175 . . . . . . . 8 𝑥 ∈ V
9 vex 3175 . . . . . . . 8 𝑦 ∈ V
108, 9opnzi 4863 . . . . . . 7 𝑥, 𝑦⟩ ≠ ∅
11 nesym 2837 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ≠ ∅ ↔ ¬ ∅ = ⟨𝑥, 𝑦⟩)
12 pm2.21 118 . . . . . . . 8 (¬ ∅ = ⟨𝑥, 𝑦⟩ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
1311, 12sylbi 205 . . . . . . 7 (⟨𝑥, 𝑦⟩ ≠ ∅ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
1410, 13ax-mp 5 . . . . . 6 (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
1514adantr 479 . . . . 5 ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
167, 15exlimi 2072 . . . 4 (∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
174, 16exlimi 2072 . . 3 (∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
181, 17sylbi 205 . 2 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19 id 22 . 2 (¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2018, 19pm2.61i 174 1 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1976  wne 2779  c0 3873  cop 4130  {copab 4636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638
This theorem is referenced by:  brabv  6575  bj-0nelmpt  32046
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