Theorem opprc1 1905

Description: Expansion of an ordered pair when the first member is a proper class.

Assertion

Ref	Expression
opprc1	⊢ (¬ A ∈ V → ⟨A, B⟩ = {∅, {B}})

Proof of Theorem opprc1

Step	Hyp	Ref	Expression
1		snprc 1838	. . . 4 ⊢ (¬ A ∈ V ↔ {A} = ∅)
2		preq1 1870	. . . 4 ⊢ ({A} = ∅ → {{A}, {A, B}} = {∅, {A, B}})
3	1, 2	sylbi 174	. . 3 ⊢ (¬ A ∈ V → {{A}, {A, B}} = {∅, {A, B}})
4		prprc 1839	. . . 4 ⊢ (¬ A ∈ V → {A, B} = {B})
5		preq2 1871	. . . 4 ⊢ ({A, B} = {B} → {∅, {A, B}} = {∅, {B}})
6	4, 5	syl 12	. . 3 ⊢ (¬ A ∈ V → {∅, {A, B}} = {∅, {B}})
7	3, 6	eqtrd 1128	. 2 ⊢ (¬ A ∈ V → {{A}, {A, B}} = {∅, {B}})
8		df-op 1815	. 2 ⊢ ⟨A, B⟩ = {{A}, {A, B}}
9	7, 8	syl5eq 1136	1 ⊢ (¬ A ∈ V → ⟨A, B⟩ = {∅, {B}})

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808  {cpr 1809  ⟨cop 1810

This theorem is referenced by:  opprc1b 1906  opprc3 1908  opth2 1909

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-16 922  ax-17 925  ax-ext 1074

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-nul 1708  df-sn 1811  df-pr 1812  df-op 1815

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