Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wopprc Structured version   Visualization version   GIF version

Theorem wopprc 37423
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 4188 . . . . . . . . 9 {∅} = {∅, ∅}
2 id 22 . . . . . . . . 9 ({∅} = {{𝐴}, ∅} → {∅} = {{𝐴}, ∅})
31, 2syl5reqr 2670 . . . . . . . 8 ({∅} = {{𝐴}, ∅} → {{𝐴}, ∅} = {∅, ∅})
4 snex 4906 . . . . . . . . 9 {𝐴} ∈ V
5 0ex 4788 . . . . . . . . 9 ∅ ∈ V
64, 5preqr1 4377 . . . . . . . 8 ({{𝐴}, ∅} = {∅, ∅} → {𝐴} = ∅)
73, 6syl 17 . . . . . . 7 ({∅} = {{𝐴}, ∅} → {𝐴} = ∅)
8 snprc 4251 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
97, 8sylibr 224 . . . . . 6 ({∅} = {{𝐴}, ∅} → ¬ 𝐴 ∈ V)
108biimpi 206 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1110preq1d 4272 . . . . . . 7 𝐴 ∈ V → {{𝐴}, ∅} = {∅, ∅})
1211, 1syl6reqr 2674 . . . . . 6 𝐴 ∈ V → {∅} = {{𝐴}, ∅})
139, 12impbii 199 . . . . 5 ({∅} = {{𝐴}, ∅} ↔ ¬ 𝐴 ∈ V)
1413con2bii 347 . . . 4 (𝐴 ∈ V ↔ ¬ {∅} = {{𝐴}, ∅})
15 snprc 4251 . . . . . . 7 𝐵 ∈ V ↔ {𝐵} = ∅)
16 eqcom 2628 . . . . . . 7 ({𝐵} = ∅ ↔ ∅ = {𝐵})
1715, 16bitr2i 265 . . . . . 6 (∅ = {𝐵} ↔ ¬ 𝐵 ∈ V)
1817con2bii 347 . . . . 5 (𝐵 ∈ V ↔ ¬ ∅ = {𝐵})
195sneqr 4369 . . . . . 6 ({∅} = {{𝐵}} → ∅ = {𝐵})
20 sneq 4185 . . . . . 6 (∅ = {𝐵} → {∅} = {{𝐵}})
2119, 20impbii 199 . . . . 5 ({∅} = {{𝐵}} ↔ ∅ = {𝐵})
2218, 21xchbinxr 325 . . . 4 (𝐵 ∈ V ↔ ¬ {∅} = {{𝐵}})
2314, 22anbi12i 733 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}))
24 pm4.56 516 . . . 4 ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
25 snex 4906 . . . . 5 {∅} ∈ V
2625elpr 4196 . . . 4 ({∅} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
2724, 26xchbinxr 325 . . 3 ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
2823, 27bitri 264 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
29 df1o2 7569 . . 3 1𝑜 = {∅}
3029eleq1i 2691 . 2 (1𝑜 ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
3128, 30xchbinxr 325 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1𝑜 ∈ {{{𝐴}, ∅}, {{𝐵}}})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 383  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  c0 3913  {csn 4175  {cpr 4177  1𝑜c1o 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200  df-dif 3575  df-un 3577  df-nul 3914  df-sn 4176  df-pr 4178  df-suc 5727  df-1o 7557
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator