Theorem 0nelop 5481

Description: A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
0nelop	⊢ ¬ ∅ ∈ ⟨𝐴, 𝐵⟩

Proof of Theorem 0nelop

Step	Hyp	Ref	Expression
1		id 23	. . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ∈ ⟨𝐴, 𝐵⟩)
2		oprcl 4866	. . . . 5 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		dfopg 4838	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4	2, 3	syl 18	. . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
5	1, 4	eleqtrd 2867	. . 3 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ∈ {{𝐴}, {𝐴, 𝐵}})
6		elpri 4615	. . 3 ⊢ (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
7	5, 6	syl 18	. 2 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
8	2	simpld 500	. . . . . 6 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → 𝐴 ∈ V)
9	8	snn0d 4743	. . . . 5 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴} ≠ ∅)
10	9	necomd 3015	. . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴})
11		prnzg 4746	. . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅)
12	8, 11	syl 18	. . . . 5 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴, 𝐵} ≠ ∅)
13	12	necomd 3015	. . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴, 𝐵})
14	10, 13	jca 521	. . 3 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}))
15		neanior 3053	. . 3 ⊢ ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
16	14, 15	sylib 221	. 2 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
17	7, 16	pm2.65i 196	1 ⊢ ¬ ∅ ∈ ⟨𝐴, 𝐵⟩

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457  ∅c0 4286  {csn 4591  {cpr 4593  ⟨cop 4597

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598

This theorem is used by:  opwo0id  5482  0nelelxp  5698