Theorem opprc1b 1906

Description: A property of an ordered pair of proper classes (due to our particular definition of ordered pair).

Assertion

Ref	Expression
opprc1b	⊢ (¬ A ∈ V ↔ ∅ ∈ ⟨A, B⟩)

Proof of Theorem opprc1b

Step	Hyp	Ref	Expression
1		0ex 1745	. . . 4 ⊢ ∅ ∈ V
2	1	pri1 1841	. . 3 ⊢ ∅ ∈ {∅, {B}}
3		opprc1 1905	. . . 4 ⊢ (¬ A ∈ V → ⟨A, B⟩ = {∅, {B}})
4	3	eleq2d 1156	. . 3 ⊢ (¬ A ∈ V → (∅ ∈ ⟨A, B⟩ ↔ ∅ ∈ {∅, {B}}))
5	2, 4	mpbiri 169	. 2 ⊢ (¬ A ∈ V → ∅ ∈ ⟨A, B⟩)
6		opeq1 1876	. . . . . 6 ⊢ (x = A → ⟨x, B⟩ = ⟨A, B⟩)
7	6	eleq2d 1156	. . . . 5 ⊢ (x = A → (∅ ∈ ⟨x, B⟩ ↔ ∅ ∈ ⟨A, B⟩))
8	7	negbid 463	. . . 4 ⊢ (x = A → (¬ ∅ ∈ ⟨x, B⟩ ↔ ¬ ∅ ∈ ⟨A, B⟩))
9		visset 1350	. . . . . . . 8 ⊢ x ∈ V
10	9	snnz 1846	. . . . . . 7 ⊢ ¬ {x} = ∅
11		cleqcom 1103	. . . . . . 7 ⊢ ({x} = ∅ ↔ ∅ = {x})
12	10, 11	mtbi 166	. . . . . 6 ⊢ ¬ ∅ = {x}
13	9	prnz 1847	. . . . . . 7 ⊢ ¬ {x, B} = ∅
14		cleqcom 1103	. . . . . . 7 ⊢ ({x, B} = ∅ ↔ ∅ = {x, B})
15	13, 14	mtbi 166	. . . . . 6 ⊢ ¬ ∅ = {x, B}
16	12, 15	pm3.2ni 440	. . . . 5 ⊢ ¬ (∅ = {x} ∨ ∅ = {x, B})
17	1	elop 1894	. . . . 5 ⊢ (∅ ∈ ⟨x, B⟩ ↔ (∅ = {x} ∨ ∅ = {x, B}))
18	16, 17	mtbir 167	. . . 4 ⊢ ¬ ∅ ∈ ⟨x, B⟩
19	8, 18	vtoclg 1383	. . 3 ⊢ (A ∈ V → ¬ ∅ ∈ ⟨A, B⟩)
20	19	con2i 89	. 2 ⊢ (∅ ∈ ⟨A, B⟩ → ¬ A ∈ V)
21	5, 20	impbi 139	1 ⊢ (¬ A ∈ V ↔ ∅ ∈ ⟨A, B⟩)

Syntax hints:  ¬ wn 1   ↔ wb 127   ∨ wo 195   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808  {cpr 1809  ⟨cop 1810

This theorem is referenced by:  opprc3 1908  opth2 1909  opelxpex 2445  onxpdisj 2476

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815

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