Theorem wopprc 44177

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		id 23	. . . . . . . . 9 ⊢ ({∅} = {{𝐴}, ∅} → {∅} = {{𝐴}, ∅})
2		dfsn2 4596	. . . . . . . . 9 ⊢ {∅} = {∅, ∅}
3	1, 2	eqtr3di 2813	. . . . . . . 8 ⊢ ({∅} = {{𝐴}, ∅} → {{𝐴}, ∅} = {∅, ∅})
4		snex 5400	. . . . . . . . 9 ⊢ {𝐴} ∈ V
5		0ex 5263	. . . . . . . . 9 ⊢ ∅ ∈ V
6	4, 5	preqr1 4805	. . . . . . . 8 ⊢ ({{𝐴}, ∅} = {∅, ∅} → {𝐴} = ∅)
7	3, 6	syl 18	. . . . . . 7 ⊢ ({∅} = {{𝐴}, ∅} → {𝐴} = ∅)
8		snprc 4677	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
9	7, 8	sylibr 237	. . . . . 6 ⊢ ({∅} = {{𝐴}, ∅} → ¬ 𝐴 ∈ V)
10	8	biimpi 219	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11	10	preq1d 4699	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {{𝐴}, ∅} = {∅, ∅})
12	2, 11	eqtr4id 2817	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {∅} = {{𝐴}, ∅})
13	9, 12	impbii 212	. . . . 5 ⊢ ({∅} = {{𝐴}, ∅} ↔ ¬ 𝐴 ∈ V)
14	13	con2bii 360	. . . 4 ⊢ (𝐴 ∈ V ↔ ¬ {∅} = {{𝐴}, ∅})
15		snprc 4677	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
16		eqcom 2770	. . . . . . 7 ⊢ ({𝐵} = ∅ ↔ ∅ = {𝐵})
17	15, 16	bitr2i 279	. . . . . 6 ⊢ (∅ = {𝐵} ↔ ¬ 𝐵 ∈ V)
18	17	con2bii 360	. . . . 5 ⊢ (𝐵 ∈ V ↔ ¬ ∅ = {𝐵})
19	5	sneqr 4797	. . . . . 6 ⊢ ({∅} = {{𝐵}} → ∅ = {𝐵})
20		sneq 4593	. . . . . 6 ⊢ (∅ = {𝐵} → {∅} = {{𝐵}})
21	19, 20	impbii 212	. . . . 5 ⊢ ({∅} = {{𝐵}} ↔ ∅ = {𝐵})
22	18, 21	xchbinxr 338	. . . 4 ⊢ (𝐵 ∈ V ↔ ¬ {∅} = {{𝐵}})
23	14, 22	anbi12i 640	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}))
24		pm4.56 1004	. . . 4 ⊢ ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
25		snex 5400	. . . . 5 ⊢ {∅} ∈ V
26	25	elpr 4608	. . . 4 ⊢ ({∅} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
27	24, 26	xchbinxr 338	. . 3 ⊢ ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
28	23, 27	bitri 278	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
29		df1o2 8494	. . 3 ⊢ 1o = {∅}
30	29	eleq1i 2854	. 2 ⊢ (1o ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
31	28, 30	xchbinxr 338	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}})

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2146  Vcvv 3453  ∅c0 4280  {csn 4587  {cpr 4589  1_oc1o 8480

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3455  df-dif 4049  df-un 4153  df-nul 4281  df-sn 4588  df-pr 4590  df-suc 6374  df-1o 8487

This theorem is used by: (None)