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Theorem 0neqopab 5492
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 3986 . . 3 (∅ {⟨x, y⟩ ∣ φ} ↔ xy(∅ = ⟨x, y φ))
3 nfopab1 3817 . . . . . 6 x{⟨x, y⟩ ∣ φ}
43nfel2 2187 . . . . 5 x {⟨x, y⟩ ∣ φ}
54nfn 1545 . . . 4 x ¬ ∅ {⟨x, y⟩ ∣ φ}
6 nfopab2 3818 . . . . . . 7 y{⟨x, y⟩ ∣ φ}
76nfel2 2187 . . . . . 6 y {⟨x, y⟩ ∣ φ}
87nfn 1545 . . . . 5 y ¬ ∅ {⟨x, y⟩ ∣ φ}
9 vex 2554 . . . . . . . 8 x V
10 vex 2554 . . . . . . . 8 y V
119, 10opnzi 3963 . . . . . . 7 x, y⟩ ≠ ∅
12 nesym 2244 . . . . . . . 8 (⟨x, y⟩ ≠ ∅ ↔ ¬ ∅ = ⟨x, y⟩)
13 pm2.21 547 . . . . . . . 8 (¬ ∅ = ⟨x, y⟩ → (∅ = ⟨x, y⟩ → ¬ ∅ {⟨x, y⟩ ∣ φ}))
1412, 13sylbi 114 . . . . . . 7 (⟨x, y⟩ ≠ ∅ → (∅ = ⟨x, y⟩ → ¬ ∅ {⟨x, y⟩ ∣ φ}))
1511, 14ax-mp 7 . . . . . 6 (∅ = ⟨x, y⟩ → ¬ ∅ {⟨x, y⟩ ∣ φ})
1615adantr 261 . . . . 5 ((∅ = ⟨x, y φ) → ¬ ∅ {⟨x, y⟩ ∣ φ})
178, 16exlimi 1482 . . . 4 (y(∅ = ⟨x, y φ) → ¬ ∅ {⟨x, y⟩ ∣ φ})
185, 17exlimi 1482 . . 3 (xy(∅ = ⟨x, y φ) → ¬ ∅ {⟨x, y⟩ ∣ φ})
192, 18sylbi 114 . 2 (∅ {⟨x, y⟩ ∣ φ} → ¬ ∅ {⟨x, y⟩ ∣ φ})
201, 19pm2.65i 567 1 ¬ ∅ {⟨x, y⟩ ∣ φ}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  wne 2201  c0 3218  cop 3370  {copab 3808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810
This theorem is referenced by: (None)
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