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Theorem spr0nelg 43515
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 975 . . . . . 6 (¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
21bicomi 225 . . . . 5 ((¬ 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
32albii 1811 . . . 4 (∀𝑝𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
4 alnex 1773 . . . 4 (∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
53, 4bitri 276 . . 3 (∀𝑝𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
6 vex 3495 . . . . . . . . 9 𝑎 ∈ V
76prnz 4704 . . . . . . . 8 {𝑎, 𝑏} ≠ ∅
87nesymi 3070 . . . . . . 7 ¬ ∅ = {𝑎, 𝑏}
9 eqeq1 2822 . . . . . . 7 (𝑝 = ∅ → (𝑝 = {𝑎, 𝑏} ↔ ∅ = {𝑎, 𝑏}))
108, 9mtbiri 328 . . . . . 6 (𝑝 = ∅ → ¬ 𝑝 = {𝑎, 𝑏})
1110alrimivv 1920 . . . . 5 (𝑝 = ∅ → ∀𝑎𝑏 ¬ 𝑝 = {𝑎, 𝑏})
12 2nexaln 1821 . . . . 5 (¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏} ↔ ∀𝑎𝑏 ¬ 𝑝 = {𝑎, 𝑏})
1311, 12sylibr 235 . . . 4 (𝑝 = ∅ → ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
1413imori 848 . . 3 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
155, 14mpgbi 1790 . 2 ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
16 df-nel 3121 . . 3 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
17 clelab 2955 . . 3 (∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
1816, 17xchbinx 335 . 2 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
1915, 18mpbir 232 1 ∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wo 841  wal 1526   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wnel 3120  c0 4288  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-v 3494  df-dif 3936  df-un 3938  df-nul 4289  df-sn 4558  df-pr 4560
This theorem is referenced by:  spr0el  43521
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