Theorem opprc2 1907

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

Assertion

Ref	Expression
opprc2	⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)

Proof of Theorem opprc2

Step	Hyp	Ref	Expression
1		prprc 1839	. . . 4 ⊢ (¬ B ∈ V → {B, A} = {A})
2		prcom 1840	. . . 4 ⊢ {B, A} = {A, B}
3		dfsn2 1819	. . . 4 ⊢ {A} = {A, A}
4	1, 2, 3	3eqtr3g 1146	. . 3 ⊢ (¬ B ∈ V → {A, B} = {A, A})
5		preq2 1871	. . 3 ⊢ ({A, B} = {A, A} → {{A}, {A, B}} = {{A}, {A, A}})
6	4, 5	syl 12	. 2 ⊢ (¬ B ∈ V → {{A}, {A, B}} = {{A}, {A, A}})
7		df-op 1815	. 2 ⊢ ⟨A, B⟩ = {{A}, {A, B}}
8		df-op 1815	. 2 ⊢ ⟨A, A⟩ = {{A}, {A, A}}
9	6, 7, 8	3eqtr4g 1147	1 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)

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

This theorem is referenced by:  opprc3 1908  brprc 2097  relsn 2485  opeldm 2534  dmsnop 2547  oprprc2 3020

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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