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Theorem spr0nelg 42051
 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 508 . . . . . 6 (¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
21bicomi 214 . . . . 5 ((¬ 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
32albii 1787 . . . 4 (∀𝑝𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
4 alnex 1746 . . . 4 (∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
53, 4bitri 264 . . 3 (∀𝑝𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
6 vex 3234 . . . . . . . . 9 𝑎 ∈ V
76prnz 4341 . . . . . . . 8 {𝑎, 𝑏} ≠ ∅
87nesymi 2880 . . . . . . 7 ¬ ∅ = {𝑎, 𝑏}
9 eqeq1 2655 . . . . . . 7 (𝑝 = ∅ → (𝑝 = {𝑎, 𝑏} ↔ ∅ = {𝑎, 𝑏}))
108, 9mtbiri 316 . . . . . 6 (𝑝 = ∅ → ¬ 𝑝 = {𝑎, 𝑏})
1110alrimivv 1896 . . . . 5 (𝑝 = ∅ → ∀𝑎𝑏 ¬ 𝑝 = {𝑎, 𝑏})
12 2nexaln 1797 . . . . 5 (¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏} ↔ ∀𝑎𝑏 ¬ 𝑝 = {𝑎, 𝑏})
1311, 12sylibr 224 . . . 4 (𝑝 = ∅ → ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
1413imori 428 . . 3 𝑝 = ∅ ∨ ¬ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
155, 14mpgbi 1765 . 2 ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
16 df-nel 2927 . . 3 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
17 clelab 2777 . . 3 (∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
1816, 17xchbinx 323 . 2 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
1915, 18mpbir 221 1 ∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637   ∉ wnel 2926  ∅c0 3948  {cpr 4212 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213 This theorem is referenced by:  spr0el  42057
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