Theorem wopprc 43214

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 22	. . . . . . . . 9 ⊢ ({∅} = {{𝐴}, ∅} → {∅} = {{𝐴}, ∅})
2		dfsn2 4591	. . . . . . . . 9 ⊢ {∅} = {∅, ∅}
3	1, 2	eqtr3di 2784	. . . . . . . 8 ⊢ ({∅} = {{𝐴}, ∅} → {{𝐴}, ∅} = {∅, ∅})
4		snex 5379	. . . . . . . . 9 ⊢ {𝐴} ∈ V
5		0ex 5250	. . . . . . . . 9 ⊢ ∅ ∈ V
6	4, 5	preqr1 4802	. . . . . . . 8 ⊢ ({{𝐴}, ∅} = {∅, ∅} → {𝐴} = ∅)
7	3, 6	syl 17	. . . . . . 7 ⊢ ({∅} = {{𝐴}, ∅} → {𝐴} = ∅)
8		snprc 4672	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
9	7, 8	sylibr 234	. . . . . 6 ⊢ ({∅} = {{𝐴}, ∅} → ¬ 𝐴 ∈ V)
10	8	biimpi 216	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11	10	preq1d 4694	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {{𝐴}, ∅} = {∅, ∅})
12	2, 11	eqtr4id 2788	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {∅} = {{𝐴}, ∅})
13	9, 12	impbii 209	. . . . 5 ⊢ ({∅} = {{𝐴}, ∅} ↔ ¬ 𝐴 ∈ V)
14	13	con2bii 357	. . . 4 ⊢ (𝐴 ∈ V ↔ ¬ {∅} = {{𝐴}, ∅})
15		snprc 4672	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
16		eqcom 2741	. . . . . . 7 ⊢ ({𝐵} = ∅ ↔ ∅ = {𝐵})
17	15, 16	bitr2i 276	. . . . . 6 ⊢ (∅ = {𝐵} ↔ ¬ 𝐵 ∈ V)
18	17	con2bii 357	. . . . 5 ⊢ (𝐵 ∈ V ↔ ¬ ∅ = {𝐵})
19	5	sneqr 4794	. . . . . 6 ⊢ ({∅} = {{𝐵}} → ∅ = {𝐵})
20		sneq 4588	. . . . . 6 ⊢ (∅ = {𝐵} → {∅} = {{𝐵}})
21	19, 20	impbii 209	. . . . 5 ⊢ ({∅} = {{𝐵}} ↔ ∅ = {𝐵})
22	18, 21	xchbinxr 335	. . . 4 ⊢ (𝐵 ∈ V ↔ ¬ {∅} = {{𝐵}})
23	14, 22	anbi12i 628	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}))
24		pm4.56 990	. . . 4 ⊢ ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
25		snex 5379	. . . . 5 ⊢ {∅} ∈ V
26	25	elpr 4603	. . . 4 ⊢ ({∅} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ ({∅} = {{𝐴}, ∅} ∨ {∅} = {{𝐵}}))
27	24, 26	xchbinxr 335	. . 3 ⊢ ((¬ {∅} = {{𝐴}, ∅} ∧ ¬ {∅} = {{𝐵}}) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
28	23, 27	bitri 275	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
29		df1o2 8402	. . 3 ⊢ 1o = {∅}
30	29	eleq1i 2825	. 2 ⊢ (1o ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {∅} ∈ {{{𝐴}, ∅}, {{𝐵}}})
31	28, 30	xchbinxr 335	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}})

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ∅c0 4283  {csn 4578  {cpr 4580  1_oc1o 8388

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-dif 3902  df-un 3904  df-nul 4284  df-sn 4579  df-pr 4581  df-suc 6321  df-1o 8395

This theorem is referenced by: (None)