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Theorem 0nelop 5351
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 4791 . . . . 5 (∅ ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 dfopg 4761 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
42, 3syl 17 . . . 4 (∅ ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
51, 4eleqtrd 2892 . . 3 (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ∈ {{𝐴}, {𝐴, 𝐵}})
6 elpri 4547 . . 3 (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
75, 6syl 17 . 2 (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
82simpld 498 . . . . . 6 (∅ ∈ ⟨𝐴, 𝐵⟩ → 𝐴 ∈ V)
9 snnzg 4670 . . . . . 6 (𝐴 ∈ V → {𝐴} ≠ ∅)
108, 9syl 17 . . . . 5 (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴} ≠ ∅)
1110necomd 3042 . . . 4 (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴})
12 prnzg 4674 . . . . . 6 (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅)
138, 12syl 17 . . . . 5 (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴, 𝐵} ≠ ∅)
1413necomd 3042 . . . 4 (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴, 𝐵})
1511, 14jca 515 . . 3 (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}))
16 neanior 3079 . . 3 ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
1715, 16sylib 221 . 2 (∅ ∈ ⟨𝐴, 𝐵⟩ → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵}))
187, 17pm2.65i 197 1 ¬ ∅ ∈ ⟨𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399  wo 844   = wceq 1538  wcel 2111  wne 2987  Vcvv 3441  c0 4243  {csn 4525  {cpr 4527  cop 4531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532
This theorem is referenced by:  opwo0id  5352  0nelelxp  5554
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