Theorem wopprc 39647

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

Step	Hyp	Ref	Expression
1		dfsn2 4580	. . . . . . . . 9 ⊢ {∅} = {∅, ∅}
2		id 22	. . . . . . . . 9 ⊢ ({∅} = {{𝐴}, ∅} → {∅} = {{𝐴}, ∅})
3	1, 2	syl5reqr 2871	. . . . . . . 8 ⊢ ({∅} = {{𝐴}, ∅} → {{𝐴}, ∅} = {∅, ∅})
4		snex 5332	. . . . . . . . 9 ⊢ {𝐴} ∈ V
5		0ex 5211	. . . . . . . . 9 ⊢ ∅ ∈ V
6	4, 5	preqr1 4779	. . . . . . . 8 ⊢ ({{𝐴}, ∅} = {∅, ∅} → {𝐴} = ∅)
7	3, 6	syl 17	. . . . . . 7 ⊢ ({∅} = {{𝐴}, ∅} → {𝐴} = ∅)
8		snprc 4653	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
9	7, 8	sylibr 236	. . . . . 6 ⊢ ({∅} = {{𝐴}, ∅} → ¬ 𝐴 ∈ V)
10	8	biimpi 218	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11	10	preq1d 4675	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {{𝐴}, ∅} = {∅, ∅})
12	11, 1	syl6reqr 2875	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {∅} = {{𝐴}, ∅})
13	9, 12	impbii 211	. . . . 5 ⊢ ({∅} = {{𝐴}, ∅} ↔ ¬ 𝐴 ∈ V)
14	13	con2bii 360	. . . 4 ⊢ (𝐴 ∈ V ↔ ¬ {∅} = {{𝐴}, ∅})
15		snprc 4653	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
16		eqcom 2828	. . . . . . 7 ⊢ ({𝐵} = ∅ ↔ ∅ = {𝐵})
17	15, 16	bitr2i 278	. . . . . 6 ⊢ (∅ = {𝐵} ↔ ¬ 𝐵 ∈ V)
18	17	con2bii 360	. . . . 5 ⊢ (𝐵 ∈ V ↔ ¬ ∅ = {𝐵})
19	5	sneqr 4771	. . . . . 6 ⊢ ({∅} = {{𝐵}} → ∅ = {𝐵})
20		sneq 4577	. . . . . 6 ⊢ (∅ = {𝐵} → {∅} = {{𝐵}})
21	19, 20	impbii 211	. . . . 5 ⊢ ({∅} = {{𝐵}} ↔ ∅ = {𝐵})
22	18, 21	xchbinxr 337	. . . 4 ⊢ (𝐵 ∈ V ↔ ¬ {∅} = {{𝐵}})
23	14, 22	anbi12i 628	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}))
24		pm4.56 985	. . . 4 ⊢ ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
25		snex 5332	. . . . 5 ⊢ {∅} ∈ V
26	25	elpr 4590	. . . 4 ⊢ ({∅} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
27	24, 26	xchbinxr 337	. . 3 ⊢ ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
28	23, 27	bitri 277	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
29		df1o2 8116	. . 3 ⊢ 1o = {∅}
30	29	eleq1i 2903	. 2 ⊢ (1o ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
31	28, 30	xchbinxr 337	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}})

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ∅c0 4291  {csn 4567  {cpr 4569  1_oc1o 8095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-un 3941  df-nul 4292  df-sn 4568  df-pr 4570  df-suc 6197  df-1o 8102

This theorem is referenced by: (None)