Theorem dmsnsnsn 2548

Description: The domain of the singleton of the singleton of a singleton.

Assertion

Ref	Expression
dmsnsnsn	⊢ dom {{{A}}} = {A}

Proof of Theorem dmsnsnsn

Step	Hyp	Ref	Expression
1		dfsn2 1819	. . . . . 6 ⊢ {A} = {A, A}
2		preq2 1871	. . . . . 6 ⊢ ({A} = {A, A} → {{A}, {A}} = {{A}, {A, A}})
3	1, 2	ax-mp 6	. . . . 5 ⊢ {{A}, {A}} = {{A}, {A, A}}
4		dfsn2 1819	. . . . 5 ⊢ {{A}} = {{A}, {A}}
5		df-op 1815	. . . . 5 ⊢ ⟨A, A⟩ = {{A}, {A, A}}
6	3, 4, 5	3eqtr4r 1127	. . . 4 ⊢ ⟨A, A⟩ = {{A}}
7	6	sneqi 1817	. . 3 ⊢ {⟨A, A⟩} = {{{A}}}
8	7	dmeqi 2532	. 2 ⊢ dom {⟨A, A⟩} = dom {{{A}}}
9		dmsnop 2547	. 2 ⊢ dom {⟨A, A⟩} = {A}
10	8, 9	eqtr3 1121	1 ⊢ dom {{{A}}} = {A}

Syntax hints:   = wceq 1091  {csn 1808  {cpr 1809  ⟨cop 1810  dom cdm 2410

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  df-br 2063  df-dm 2428

metamath.org