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Theorem spr0nelg 44880
Description: The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
Assertion
Ref Expression
spr0nelg ∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Distinct variable groups:   𝑝,𝑎   𝑝,𝑏

Proof of Theorem spr0nelg
StepHypRef Expression
1 ianor 978 . . . . . 6 (¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
21bicomi 223 . . . . 5 ((¬ 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
32albii 1825 . . . 4 (∀𝑝𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
4 alnex 1787 . . . 4 (∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
53, 4bitri 274 . . 3 (∀𝑝𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
6 vex 3434 . . . . . . . . 9 𝑎 ∈ V
76prnz 4718 . . . . . . . 8 {𝑎, 𝑏} ≠ ∅
87nesymi 3002 . . . . . . 7 ¬ ∅ = {𝑎, 𝑏}
9 eqeq1 2743 . . . . . . 7 (𝑝 = ∅ → (𝑝 = {𝑎, 𝑏} ↔ ∅ = {𝑎, 𝑏}))
108, 9mtbiri 326 . . . . . 6 (𝑝 = ∅ → ¬ 𝑝 = {𝑎, 𝑏})
1110alrimivv 1934 . . . . 5 (𝑝 = ∅ → ∀𝑎𝑏 ¬ 𝑝 = {𝑎, 𝑏})
12 2nexaln 1835 . . . . 5 (¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏} ↔ ∀𝑎𝑏 ¬ 𝑝 = {𝑎, 𝑏})
1311, 12sylibr 233 . . . 4 (𝑝 = ∅ → ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
1413imori 850 . . 3 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
155, 14mpgbi 1804 . 2 ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
16 df-nel 3051 . . 3 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
17 clelab 2884 . . 3 (∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
1816, 17xchbinx 333 . 2 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
1915, 18mpbir 230 1 ∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wo 843  wal 1539   = wceq 1541  wex 1785  wcel 2109  {cab 2716  wnel 3050  c0 4261  {cpr 4568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-12 2174  ax-ext 2710
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-nel 3051  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-sn 4567  df-pr 4569
This theorem is referenced by:  spr0el  44886
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