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 39966
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 4538 . . . . . . . . 9 {∅} = {∅, ∅}
2 id 22 . . . . . . . . 9 ({∅} = {{𝐴}, ∅} → {∅} = {{𝐴}, ∅})
31, 2syl5reqr 2848 . . . . . . . 8 ({∅} = {{𝐴}, ∅} → {{𝐴}, ∅} = {∅, ∅})
4 snex 5297 . . . . . . . . 9 {𝐴} ∈ V
5 0ex 5175 . . . . . . . . 9 ∅ ∈ V
64, 5preqr1 4739 . . . . . . . 8 ({{𝐴}, ∅} = {∅, ∅} → {𝐴} = ∅)
73, 6syl 17 . . . . . . 7 ({∅} = {{𝐴}, ∅} → {𝐴} = ∅)
8 snprc 4613 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
97, 8sylibr 237 . . . . . 6 ({∅} = {{𝐴}, ∅} → ¬ 𝐴 ∈ V)
108biimpi 219 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1110preq1d 4635 . . . . . . 7 𝐴 ∈ V → {{𝐴}, ∅} = {∅, ∅})
121, 11eqtr4id 2852 . . . . . 6 𝐴 ∈ V → {∅} = {{𝐴}, ∅})
139, 12impbii 212 . . . . 5 ({∅} = {{𝐴}, ∅} ↔ ¬ 𝐴 ∈ V)
1413con2bii 361 . . . 4 (𝐴 ∈ V ↔ ¬ {∅} = {{𝐴}, ∅})
15 snprc 4613 . . . . . . 7 𝐵 ∈ V ↔ {𝐵} = ∅)
16 eqcom 2805 . . . . . . 7 ({𝐵} = ∅ ↔ ∅ = {𝐵})
1715, 16bitr2i 279 . . . . . 6 (∅ = {𝐵} ↔ ¬ 𝐵 ∈ V)
1817con2bii 361 . . . . 5 (𝐵 ∈ V ↔ ¬ ∅ = {𝐵})
195sneqr 4731 . . . . . 6 ({∅} = {{𝐵}} → ∅ = {𝐵})
20 sneq 4535 . . . . . 6 (∅ = {𝐵} → {∅} = {{𝐵}})
2119, 20impbii 212 . . . . 5 ({∅} = {{𝐵}} ↔ ∅ = {𝐵})
2218, 21xchbinxr 338 . . . 4 (𝐵 ∈ V ↔ ¬ {∅} = {{𝐵}})
2314, 22anbi12i 629 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}))
24 pm4.56 986 . . . 4 ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
25 snex 5297 . . . . 5 {∅} ∈ V
2625elpr 4548 . . . 4 ({∅} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
2724, 26xchbinxr 338 . . 3 ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
2823, 27bitri 278 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
29 df1o2 8099 . . 3 1o = {∅}
3029eleq1i 2880 . 2 (1o ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
3128, 30xchbinxr 338 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wo 844   = wceq 1538  wcel 2111  Vcvv 3441  c0 4243  {csn 4525  {cpr 4527  1oc1o 8078
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-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528  df-suc 6165  df-1o 8085
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator