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Theorem spr0nelg 45758
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 981 . . . . . 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 1822 . . . 4 (∀𝑝𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
4 alnex 1784 . . . 4 (∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
53, 4bitri 275 . . 3 (∀𝑝𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
6 vex 3451 . . . . . . . . 9 𝑎 ∈ V
76prnz 4742 . . . . . . . 8 {𝑎, 𝑏} ≠ ∅
87nesymi 2998 . . . . . . 7 ¬ ∅ = {𝑎, 𝑏}
9 eqeq1 2737 . . . . . . 7 (𝑝 = ∅ → (𝑝 = {𝑎, 𝑏} ↔ ∅ = {𝑎, 𝑏}))
108, 9mtbiri 327 . . . . . 6 (𝑝 = ∅ → ¬ 𝑝 = {𝑎, 𝑏})
1110alrimivv 1932 . . . . 5 (𝑝 = ∅ → ∀𝑎𝑏 ¬ 𝑝 = {𝑎, 𝑏})
12 2nexaln 1833 . . . . 5 (¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏} ↔ ∀𝑎𝑏 ¬ 𝑝 = {𝑎, 𝑏})
1311, 12sylibr 233 . . . 4 (𝑝 = ∅ → ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
1413imori 853 . . 3 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
155, 14mpgbi 1801 . 2 ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
16 df-nel 3047 . . 3 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
17 clelab 2880 . . 3 (∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
1816, 17xchbinx 334 . 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 397  wo 846  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wnel 3046  c0 4286  {cpr 4592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-nel 3047  df-v 3449  df-dif 3917  df-un 3919  df-nul 4287  df-sn 4591  df-pr 4593
This theorem is referenced by:  spr0el  45764
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