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Theorem wopprc 38123
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 dfsn2 4329 . . . . . . . . 9 {∅} = {∅, ∅}
2 id 22 . . . . . . . . 9 ({∅} = {{𝐴}, ∅} → {∅} = {{𝐴}, ∅})
31, 2syl5reqr 2820 . . . . . . . 8 ({∅} = {{𝐴}, ∅} → {{𝐴}, ∅} = {∅, ∅})
4 snex 5036 . . . . . . . . 9 {𝐴} ∈ V
5 0ex 4924 . . . . . . . . 9 ∅ ∈ V
64, 5preqr1 4511 . . . . . . . 8 ({{𝐴}, ∅} = {∅, ∅} → {𝐴} = ∅)
73, 6syl 17 . . . . . . 7 ({∅} = {{𝐴}, ∅} → {𝐴} = ∅)
8 snprc 4389 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
97, 8sylibr 224 . . . . . 6 ({∅} = {{𝐴}, ∅} → ¬ 𝐴 ∈ V)
108biimpi 206 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1110preq1d 4410 . . . . . . 7 𝐴 ∈ V → {{𝐴}, ∅} = {∅, ∅})
1211, 1syl6reqr 2824 . . . . . 6 𝐴 ∈ V → {∅} = {{𝐴}, ∅})
139, 12impbii 199 . . . . 5 ({∅} = {{𝐴}, ∅} ↔ ¬ 𝐴 ∈ V)
1413con2bii 346 . . . 4 (𝐴 ∈ V ↔ ¬ {∅} = {{𝐴}, ∅})
15 snprc 4389 . . . . . . 7 𝐵 ∈ V ↔ {𝐵} = ∅)
16 eqcom 2778 . . . . . . 7 ({𝐵} = ∅ ↔ ∅ = {𝐵})
1715, 16bitr2i 265 . . . . . 6 (∅ = {𝐵} ↔ ¬ 𝐵 ∈ V)
1817con2bii 346 . . . . 5 (𝐵 ∈ V ↔ ¬ ∅ = {𝐵})
195sneqr 4504 . . . . . 6 ({∅} = {{𝐵}} → ∅ = {𝐵})
20 sneq 4326 . . . . . 6 (∅ = {𝐵} → {∅} = {{𝐵}})
2119, 20impbii 199 . . . . 5 ({∅} = {{𝐵}} ↔ ∅ = {𝐵})
2218, 21xchbinxr 324 . . . 4 (𝐵 ∈ V ↔ ¬ {∅} = {{𝐵}})
2314, 22anbi12i 604 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}))
24 pm4.56 947 . . . 4 ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
25 snex 5036 . . . . 5 {∅} ∈ V
2625elpr 4338 . . . 4 ({∅} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
2724, 26xchbinxr 324 . . 3 ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
2823, 27bitri 264 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
29 df1o2 7726 . . 3 1𝑜 = {∅}
3029eleq1i 2841 . 2 (1𝑜 ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
3128, 30xchbinxr 324 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1𝑜 ∈ {{{𝐴}, ∅}, {{𝐵}}})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wo 826   = wceq 1631  wcel 2145  Vcvv 3351  c0 4063  {csn 4316  {cpr 4318  1𝑜c1o 7706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-sn 4317  df-pr 4319  df-suc 5872  df-1o 7713
This theorem is referenced by: (None)
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