Theorem opprc3 1908

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

Assertion

Ref	Expression
opprc3	⊢ ((¬ A ∈ V ∧ ¬ B ∈ V) ↔ ⟨A, B⟩ = {∅})

Proof of Theorem opprc3

Step	Hyp	Ref	Expression
1		opprc2 1907	. . . 4 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)
2		opprc1 1905	. . . . 5 ⊢ (¬ A ∈ V → ⟨A, A⟩ = {∅, {A}})
3		snprc 1838	. . . . . 6 ⊢ (¬ A ∈ V ↔ {A} = ∅)
4		preq2 1871	. . . . . 6 ⊢ ({A} = ∅ → {∅, {A}} = {∅, ∅})
5	3, 4	sylbi 174	. . . . 5 ⊢ (¬ A ∈ V → {∅, {A}} = {∅, ∅})
6	2, 5	eqtrd 1128	. . . 4 ⊢ (¬ A ∈ V → ⟨A, A⟩ = {∅, ∅})
7	1, 6	sylan9eqr 1145	. . 3 ⊢ ((¬ A ∈ V ∧ ¬ B ∈ V) → ⟨A, B⟩ = {∅, ∅})
8		dfsn2 1819	. . 3 ⊢ {∅} = {∅, ∅}
9	7, 8	syl6eqr 1142	. 2 ⊢ ((¬ A ∈ V ∧ ¬ B ∈ V) → ⟨A, B⟩ = {∅})
10		0ex 1745	. . . . . 6 ⊢ ∅ ∈ V
11	10	snid 1830	. . . . 5 ⊢ ∅ ∈ {∅}
12		eleq2 1150	. . . . 5 ⊢ (⟨A, B⟩ = {∅} → (∅ ∈ ⟨A, B⟩ ↔ ∅ ∈ {∅}))
13	11, 12	mpbiri 169	. . . 4 ⊢ (⟨A, B⟩ = {∅} → ∅ ∈ ⟨A, B⟩)
14		opprc1b 1906	. . . 4 ⊢ (¬ A ∈ V ↔ ∅ ∈ ⟨A, B⟩)
15	13, 14	sylibr 175	. . 3 ⊢ (⟨A, B⟩ = {∅} → ¬ A ∈ V)
16		opprc1 1905	. . . . . 6 ⊢ (¬ A ∈ V → ⟨A, B⟩ = {∅, {B}})
17	16	cleq1d 1109	. . . . 5 ⊢ (¬ A ∈ V → (⟨A, B⟩ = {∅} ↔ {∅, {B}} = {∅}))
18		snex 1859	. . . . . . 7 ⊢ {B} ∈ V
19	18, 10	prer2 1873	. . . . . 6 ⊢ ({∅, {B}} = {∅, ∅} → {B} = ∅)
20	8	cleq2i 1111	. . . . . 6 ⊢ ({∅, {B}} = {∅} ↔ {∅, {B}} = {∅, ∅})
21		snprc 1838	. . . . . 6 ⊢ (¬ B ∈ V ↔ {B} = ∅)
22	19, 20, 21	3imtr4 192	. . . . 5 ⊢ ({∅, {B}} = {∅} → ¬ B ∈ V)
23	17, 22	syl6bi 187	. . . 4 ⊢ (¬ A ∈ V → (⟨A, B⟩ = {∅} → ¬ B ∈ V))
24	23	anc2li 250	. . 3 ⊢ (¬ A ∈ V → (⟨A, B⟩ = {∅} → (¬ A ∈ V ∧ ¬ B ∈ V)))
25	15, 24	mpcom 49	. 2 ⊢ (⟨A, B⟩ = {∅} → (¬ A ∈ V ∧ ¬ B ∈ V))
26	9, 25	impbi 139	1 ⊢ ((¬ A ∈ V ∧ ¬ B ∈ V) ↔ ⟨A, B⟩ = {∅})

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808  {cpr 1809  ⟨cop 1810

This theorem is referenced by:  dmsnsn0 2544  dmsnop 2547

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

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-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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