Theorem spr0nelg 44928

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 979	. . . . . 6 ⊢ (¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
2	1	bicomi 223	. . . . 5 ⊢ ((¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
3	2	albii 1822	. . . 4 ⊢ (∀𝑝(¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
4		alnex 1784	. . . 4 ⊢ (∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
5	3, 4	bitri 274	. . 3 ⊢ (∀𝑝(¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
6		vex 3436	. . . . . . . . 9 ⊢ 𝑎 ∈ V
7	6	prnz 4713	. . . . . . . 8 ⊢ {𝑎, 𝑏} ≠ ∅
8	7	nesymi 3001	. . . . . . 7 ⊢ ¬ ∅ = {𝑎, 𝑏}
9		eqeq1 2742	. . . . . . 7 ⊢ (𝑝 = ∅ → (𝑝 = {𝑎, 𝑏} ↔ ∅ = {𝑎, 𝑏}))
10	8, 9	mtbiri 327	. . . . . 6 ⊢ (𝑝 = ∅ → ¬ 𝑝 = {𝑎, 𝑏})
11	10	alrimivv 1931	. . . . 5 ⊢ (𝑝 = ∅ → ∀𝑎∀𝑏 ¬ 𝑝 = {𝑎, 𝑏})
12		2nexaln 1832	. . . . 5 ⊢ (¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏} ↔ ∀𝑎∀𝑏 ¬ 𝑝 = {𝑎, 𝑏})
13	11, 12	sylibr 233	. . . 4 ⊢ (𝑝 = ∅ → ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
14	13	imori 851	. . 3 ⊢ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
15	5, 14	mpgbi 1801	. 2 ⊢ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
16		df-nel 3050	. . 3 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
17		clelab 2883	. . 3 ⊢ (∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
18	16, 17	xchbinx 334	. 2 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
19	15, 18	mpbir 230	1 ⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

Syntax hints:  ¬ wn 3   ∧ wa 396   ∨ wo 844  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ∉ wnel 3049  ∅c0 4256  {cpr 4563

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-nel 3050  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564

This theorem is referenced by:  spr0el  44934