Theorem spr0nelg 43658

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 978	. . . . . 6 ⊢ (¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
2	1	bicomi 226	. . . . 5 ⊢ ((¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
3	2	albii 1820	. . . 4 ⊢ (∀𝑝(¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
4		alnex 1782	. . . 4 ⊢ (∀𝑝 ¬ (𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
5	3, 4	bitri 277	. . 3 ⊢ (∀𝑝(¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
6		vex 3497	. . . . . . . . 9 ⊢ 𝑎 ∈ V
7	6	prnz 4712	. . . . . . . 8 ⊢ {𝑎, 𝑏} ≠ ∅
8	7	nesymi 3073	. . . . . . 7 ⊢ ¬ ∅ = {𝑎, 𝑏}
9		eqeq1 2825	. . . . . . 7 ⊢ (𝑝 = ∅ → (𝑝 = {𝑎, 𝑏} ↔ ∅ = {𝑎, 𝑏}))
10	8, 9	mtbiri 329	. . . . . 6 ⊢ (𝑝 = ∅ → ¬ 𝑝 = {𝑎, 𝑏})
11	10	alrimivv 1929	. . . . 5 ⊢ (𝑝 = ∅ → ∀𝑎∀𝑏 ¬ 𝑝 = {𝑎, 𝑏})
12		2nexaln 1830	. . . . 5 ⊢ (¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏} ↔ ∀𝑎∀𝑏 ¬ 𝑝 = {𝑎, 𝑏})
13	11, 12	sylibr 236	. . . 4 ⊢ (𝑝 = ∅ → ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
14	13	imori 850	. . 3 ⊢ (¬ 𝑝 = ∅ ∨ ¬ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
15	5, 14	mpgbi 1799	. 2 ⊢ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
16		df-nel 3124	. . 3 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
17		clelab 2958	. . 3 ⊢ (∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
18	16, 17	xchbinx 336	. 2 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∃𝑝(𝑝 = ∅ ∧ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
19	15, 18	mpbir 233	1 ⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

Syntax hints:  ¬ wn 3   ∧ wa 398   ∨ wo 843  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2799   ∉ wnel 3123  ∅c0 4291  {cpr 4569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-v 3496  df-dif 3939  df-un 3941  df-nul 4292  df-sn 4568  df-pr 4570

This theorem is referenced by:  spr0el  43664