Theorem spr0nelg 46807

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 980	. . . . . 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 1814	. . . 4 ⊢ (∀𝑝(¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
4		alnex 1776	. . . 4 ⊢ (∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
5	3, 4	bitri 275	. . 3 ⊢ (∀𝑝(¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
6		vex 3474	. . . . . . . . 9 ⊢ 𝑎 ∈ V
7	6	prnz 4778	. . . . . . . 8 ⊢ {𝑎, 𝑏} ≠ ∅
8	7	nesymi 2994	. . . . . . 7 ⊢ ¬ ∅ = {𝑎, 𝑏}
9		eqeq1 2732	. . . . . . 7 ⊢ (𝑝 = ∅ → (𝑝 = {𝑎, 𝑏} ↔ ∅ = {𝑎, 𝑏}))
10	8, 9	mtbiri 327	. . . . . 6 ⊢ (𝑝 = ∅ → ¬ 𝑝 = {𝑎, 𝑏})
11	10	alrimivv 1924	. . . . 5 ⊢ (𝑝 = ∅ → ∀𝑎∀𝑏 ¬ 𝑝 = {𝑎, 𝑏})
12		2nexaln 1825	. . . . 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 853	. . 3 ⊢ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
15	5, 14	mpgbi 1793	. 2 ⊢ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
16		df-nel 3043	. . 3 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
17		clelab 2875	. . 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 395   ∨ wo 846  ∀wal 1532   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2705   ∉ wnel 3042  ∅c0 4319  {cpr 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-nel 3043  df-v 3472  df-dif 3948  df-un 3950  df-nul 4320  df-sn 4626  df-pr 4628

This theorem is referenced by:  spr0el  46813