Theorem 0nelop 5471

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 22	. . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ∈ ⟨𝐴, 𝐵⟩)
2		oprcl 4875	. . . . 5 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		dfopg 4847	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4	2, 3	syl 17	. . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
5	1, 4	eleqtrd 2836	. . 3 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ∈ {{𝐴}, {𝐴, 𝐵}})
6		elpri 4625	. . 3 ⊢ (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
7	5, 6	syl 17	. 2 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
8	2	simpld 494	. . . . . 6 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → 𝐴 ∈ V)
9	8	snn0d 4751	. . . . 5 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴} ≠ ∅)
10	9	necomd 2987	. . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴})
11		prnzg 4754	. . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅)
12	8, 11	syl 17	. . . . 5 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴, 𝐵} ≠ ∅)
13	12	necomd 2987	. . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴, 𝐵})
14	10, 13	jca 511	. . 3 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}))
15		neanior 3025	. . 3 ⊢ ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
16	14, 15	sylib 218	. 2 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
17	7, 16	pm2.65i 194	1 ⊢ ¬ ∅ ∈ ⟨𝐴, 𝐵⟩

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  Vcvv 3459  ∅c0 4308  {csn 4601  {cpr 4603  ⟨cop 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608

This theorem is referenced by:  opwo0id  5472  0nelelxp  5689