Theorem spr0nelg 48114

Description: The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.)

Assertion

Ref	Expression
spr0nelg	⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

Distinct variable groups:   𝑝,𝑎   𝑝,𝑏

Proof of Theorem spr0nelg

Step	Hyp	Ref	Expression
1		ianor 997	. . . . . 6 ⊢ (¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
2	1	bicomi 227	. . . . 5 ⊢ ((¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
3	2	albii 1846	. . . 4 ⊢ (∀𝑝(¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
4		alnex 1808	. . . 4 ⊢ (∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
5	3, 4	bitri 278	. . 3 ⊢ (∀𝑝(¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
6		vex 3467	. . . . . . . . 9 ⊢ 𝑎 ∈ V
7	6	prnz 4748	. . . . . . . 8 ⊢ {𝑎, 𝑏} ≠ ∅
8	7	nesymi 3021	. . . . . . 7 ⊢ ¬ ∅ = {𝑎, 𝑏}
9		eqeq1 2773	. . . . . . 7 ⊢ (𝑝 = ∅ → (𝑝 = {𝑎, 𝑏} ↔ ∅ = {𝑎, 𝑏}))
10	8, 9	mtbiri 330	. . . . . 6 ⊢ (𝑝 = ∅ → ¬ 𝑝 = {𝑎, 𝑏})
11	10	alrimivv 1955	. . . . 5 ⊢ (𝑝 = ∅ → ∀𝑎∀𝑏 ¬ 𝑝 = {𝑎, 𝑏})
12		2nexaln 1857	. . . . 5 ⊢ (¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏} ↔ ∀𝑎∀𝑏 ¬ 𝑝 = {𝑎, 𝑏})
13	11, 12	sylibr 237	. . . 4 ⊢ (𝑝 = ∅ → ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
14	13	imori 867	. . 3 ⊢ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
15	5, 14	mpgbi 1825	. 2 ⊢ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
16		df-nel 3071	. . 3 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
17		clelab 2913	. . 3 ⊢ (∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
18	16, 17	xchbinx 337	. 2 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
19	15, 18	mpbir 234	1 ⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

Syntax hints:  ¬ wn 3   ∧ wa 400   ∨ wo 860  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747   ∉ wnel 3070  ∅c0 4294  {cpr 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-nel 3071  df-v 3465  df-dif 3916  df-un 3918  df-nul 4295  df-sn 4595  df-pr 4597

This theorem is referenced by:  spr0el  48120