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Theorem 0neqopab 5809
 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 id 19 . 2 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2 elopab 4175 . . 3 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3 nfopab1 3992 . . . . . 6 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
43nfel2 2292 . . . . 5 𝑥∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
54nfn 1636 . . . 4 𝑥 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 nfopab2 3993 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
76nfel2 2292 . . . . . 6 𝑦∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
87nfn 1636 . . . . 5 𝑦 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
9 vex 2684 . . . . . . . 8 𝑥 ∈ V
10 vex 2684 . . . . . . . 8 𝑦 ∈ V
119, 10opnzi 4152 . . . . . . 7 𝑥, 𝑦⟩ ≠ ∅
12 nesym 2351 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ≠ ∅ ↔ ¬ ∅ = ⟨𝑥, 𝑦⟩)
13 pm2.21 606 . . . . . . . 8 (¬ ∅ = ⟨𝑥, 𝑦⟩ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
1412, 13sylbi 120 . . . . . . 7 (⟨𝑥, 𝑦⟩ ≠ ∅ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
1511, 14ax-mp 5 . . . . . 6 (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
1615adantr 274 . . . . 5 ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
178, 16exlimi 1573 . . . 4 (∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
185, 17exlimi 1573 . . 3 (∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
192, 18sylbi 120 . 2 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
201, 19pm2.65i 628 1 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ≠ wne 2306  ∅c0 3358  ⟨cop 3525  {copab 3983 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985 This theorem is referenced by: (None)
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