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Theorem spr0nelg 43007
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 964 . . . . . 6 (¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
21bicomi 216 . . . . 5 ((¬ 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
32albii 1782 . . . 4 (∀𝑝𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
4 alnex 1744 . . . 4 (∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
53, 4bitri 267 . . 3 (∀𝑝𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
6 vex 3418 . . . . . . . . 9 𝑎 ∈ V
76prnz 4587 . . . . . . . 8 {𝑎, 𝑏} ≠ ∅
87nesymi 3024 . . . . . . 7 ¬ ∅ = {𝑎, 𝑏}
9 eqeq1 2782 . . . . . . 7 (𝑝 = ∅ → (𝑝 = {𝑎, 𝑏} ↔ ∅ = {𝑎, 𝑏}))
108, 9mtbiri 319 . . . . . 6 (𝑝 = ∅ → ¬ 𝑝 = {𝑎, 𝑏})
1110alrimivv 1887 . . . . 5 (𝑝 = ∅ → ∀𝑎𝑏 ¬ 𝑝 = {𝑎, 𝑏})
12 2nexaln 1792 . . . . 5 (¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏} ↔ ∀𝑎𝑏 ¬ 𝑝 = {𝑎, 𝑏})
1311, 12sylibr 226 . . . 4 (𝑝 = ∅ → ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
1413imori 840 . . 3 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
155, 14mpgbi 1761 . 2 ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
16 df-nel 3074 . . 3 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
17 clelab 2913 . . 3 (∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
1816, 17xchbinx 326 . 2 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
1915, 18mpbir 223 1 ∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 387  wo 833  wal 1505   = wceq 1507  wex 1742  wcel 2050  {cab 2758  wnel 3073  c0 4180  {cpr 4444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-v 3417  df-dif 3834  df-un 3836  df-nul 4181  df-sn 4443  df-pr 4445
This theorem is referenced by:  spr0el  43013
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