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Theorem wopprc 39057
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 4449 . . . . . . . . 9 {∅} = {∅, ∅}
2 id 22 . . . . . . . . 9 ({∅} = {{𝐴}, ∅} → {∅} = {{𝐴}, ∅})
31, 2syl5reqr 2824 . . . . . . . 8 ({∅} = {{𝐴}, ∅} → {{𝐴}, ∅} = {∅, ∅})
4 snex 5185 . . . . . . . . 9 {𝐴} ∈ V
5 0ex 5065 . . . . . . . . 9 ∅ ∈ V
64, 5preqr1 4650 . . . . . . . 8 ({{𝐴}, ∅} = {∅, ∅} → {𝐴} = ∅)
73, 6syl 17 . . . . . . 7 ({∅} = {{𝐴}, ∅} → {𝐴} = ∅)
8 snprc 4524 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
97, 8sylibr 226 . . . . . 6 ({∅} = {{𝐴}, ∅} → ¬ 𝐴 ∈ V)
108biimpi 208 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1110preq1d 4546 . . . . . . 7 𝐴 ∈ V → {{𝐴}, ∅} = {∅, ∅})
1211, 1syl6reqr 2828 . . . . . 6 𝐴 ∈ V → {∅} = {{𝐴}, ∅})
139, 12impbii 201 . . . . 5 ({∅} = {{𝐴}, ∅} ↔ ¬ 𝐴 ∈ V)
1413con2bii 350 . . . 4 (𝐴 ∈ V ↔ ¬ {∅} = {{𝐴}, ∅})
15 snprc 4524 . . . . . . 7 𝐵 ∈ V ↔ {𝐵} = ∅)
16 eqcom 2780 . . . . . . 7 ({𝐵} = ∅ ↔ ∅ = {𝐵})
1715, 16bitr2i 268 . . . . . 6 (∅ = {𝐵} ↔ ¬ 𝐵 ∈ V)
1817con2bii 350 . . . . 5 (𝐵 ∈ V ↔ ¬ ∅ = {𝐵})
195sneqr 4642 . . . . . 6 ({∅} = {{𝐵}} → ∅ = {𝐵})
20 sneq 4446 . . . . . 6 (∅ = {𝐵} → {∅} = {{𝐵}})
2119, 20impbii 201 . . . . 5 ({∅} = {{𝐵}} ↔ ∅ = {𝐵})
2218, 21xchbinxr 327 . . . 4 (𝐵 ∈ V ↔ ¬ {∅} = {{𝐵}})
2314, 22anbi12i 618 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}))
24 pm4.56 972 . . . 4 ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
25 snex 5185 . . . . 5 {∅} ∈ V
2625elpr 4459 . . . 4 ({∅} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
2724, 26xchbinxr 327 . . 3 ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
2823, 27bitri 267 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
29 df1o2 7917 . . 3 1o = {∅}
3029eleq1i 2851 . 2 (1o ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
3128, 30xchbinxr 327 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wa 387  wo 834   = wceq 1508  wcel 2051  Vcvv 3410  c0 4173  {csn 4436  {cpr 4438  1oc1o 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-v 3412  df-dif 3827  df-un 3829  df-nul 4174  df-sn 4437  df-pr 4439  df-suc 6033  df-1o 7904
This theorem is referenced by: (None)
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