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Theorem wopprc 43019
Description: Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)
Assertion
Ref Expression
wopprc ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}})

Proof of Theorem wopprc
StepHypRef Expression
1 id 22 . . . . . . . . 9 ({∅} = {{𝐴}, ∅} → {∅} = {{𝐴}, ∅})
2 dfsn2 4644 . . . . . . . . 9 {∅} = {∅, ∅}
31, 2eqtr3di 2790 . . . . . . . 8 ({∅} = {{𝐴}, ∅} → {{𝐴}, ∅} = {∅, ∅})
4 snex 5442 . . . . . . . . 9 {𝐴} ∈ V
5 0ex 5313 . . . . . . . . 9 ∅ ∈ V
64, 5preqr1 4853 . . . . . . . 8 ({{𝐴}, ∅} = {∅, ∅} → {𝐴} = ∅)
73, 6syl 17 . . . . . . 7 ({∅} = {{𝐴}, ∅} → {𝐴} = ∅)
8 snprc 4722 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
97, 8sylibr 234 . . . . . 6 ({∅} = {{𝐴}, ∅} → ¬ 𝐴 ∈ V)
108biimpi 216 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1110preq1d 4744 . . . . . . 7 𝐴 ∈ V → {{𝐴}, ∅} = {∅, ∅})
122, 11eqtr4id 2794 . . . . . 6 𝐴 ∈ V → {∅} = {{𝐴}, ∅})
139, 12impbii 209 . . . . 5 ({∅} = {{𝐴}, ∅} ↔ ¬ 𝐴 ∈ V)
1413con2bii 357 . . . 4 (𝐴 ∈ V ↔ ¬ {∅} = {{𝐴}, ∅})
15 snprc 4722 . . . . . . 7 𝐵 ∈ V ↔ {𝐵} = ∅)
16 eqcom 2742 . . . . . . 7 ({𝐵} = ∅ ↔ ∅ = {𝐵})
1715, 16bitr2i 276 . . . . . 6 (∅ = {𝐵} ↔ ¬ 𝐵 ∈ V)
1817con2bii 357 . . . . 5 (𝐵 ∈ V ↔ ¬ ∅ = {𝐵})
195sneqr 4845 . . . . . 6 ({∅} = {{𝐵}} → ∅ = {𝐵})
20 sneq 4641 . . . . . 6 (∅ = {𝐵} → {∅} = {{𝐵}})
2119, 20impbii 209 . . . . 5 ({∅} = {{𝐵}} ↔ ∅ = {𝐵})
2218, 21xchbinxr 335 . . . 4 (𝐵 ∈ V ↔ ¬ {∅} = {{𝐵}})
2314, 22anbi12i 628 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}))
24 pm4.56 990 . . . 4 ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
25 snex 5442 . . . . 5 {∅} ∈ V
2625elpr 4655 . . . 4 ({∅} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
2724, 26xchbinxr 335 . . 3 ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
2823, 27bitri 275 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
29 df1o2 8512 . . 3 1o = {∅}
3029eleq1i 2830 . 2 (1o ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
3128, 30xchbinxr 335 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {csn 4631  {cpr 4633  1oc1o 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-pr 4634  df-suc 6392  df-1o 8505
This theorem is referenced by: (None)
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