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Theorem 0nelop 5242
Description: A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
0nelop ¬ ∅ ∈ ⟨𝐴, 𝐵

Proof of Theorem 0nelop
StepHypRef Expression
1 id 22 . . . 4 (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ∈ ⟨𝐴, 𝐵⟩)
2 oprcl 4703 . . . . 5 (∅ ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 dfopg 4675 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
42, 3syl 17 . . . 4 (∅ ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
51, 4eleqtrd 2869 . . 3 (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ∈ {{𝐴}, {𝐴, 𝐵}})
6 elpri 4463 . . 3 (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
75, 6syl 17 . 2 (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
82simpld 487 . . . . . 6 (∅ ∈ ⟨𝐴, 𝐵⟩ → 𝐴 ∈ V)
9 snnzg 4584 . . . . . 6 (𝐴 ∈ V → {𝐴} ≠ ∅)
108, 9syl 17 . . . . 5 (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴} ≠ ∅)
1110necomd 3023 . . . 4 (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴})
12 prnzg 4587 . . . . . 6 (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅)
138, 12syl 17 . . . . 5 (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴, 𝐵} ≠ ∅)
1413necomd 3023 . . . 4 (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴, 𝐵})
1511, 14jca 504 . . 3 (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}))
16 neanior 3061 . . 3 ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
1715, 16sylib 210 . 2 (∅ ∈ ⟨𝐴, 𝐵⟩ → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
187, 17pm2.65i 186 1 ¬ ∅ ∈ ⟨𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 387  wo 833   = wceq 1507  wcel 2050  wne 2968  Vcvv 3416  c0 4179  {csn 4441  {cpr 4443  cop 4447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448
This theorem is referenced by:  opwo0id  5243  0nelelxp  5442
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