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Theorem 0nelop 5501
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 4899 . . . . 5 (∅ ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 dfopg 4871 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
42, 3syl 17 . . . 4 (∅ ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
51, 4eleqtrd 2843 . . 3 (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ∈ {{𝐴}, {𝐴, 𝐵}})
6 elpri 4649 . . 3 (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
75, 6syl 17 . 2 (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
82simpld 494 . . . . . 6 (∅ ∈ ⟨𝐴, 𝐵⟩ → 𝐴 ∈ V)
98snn0d 4775 . . . . 5 (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴} ≠ ∅)
109necomd 2996 . . . 4 (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴})
11 prnzg 4778 . . . . . 6 (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅)
128, 11syl 17 . . . . 5 (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴, 𝐵} ≠ ∅)
1312necomd 2996 . . . 4 (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴, 𝐵})
1410, 13jca 511 . . 3 (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}))
15 neanior 3035 . . 3 ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
1614, 15sylib 218 . 2 (∅ ∈ ⟨𝐴, 𝐵⟩ → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
177, 16pm2.65i 194 1 ¬ ∅ ∈ ⟨𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  c0 4333  {csn 4626  {cpr 4628  cop 4632
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 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633
This theorem is referenced by:  opwo0id  5502  0nelelxp  5720
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