Theorem dmsnsnsn 3325

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 2417	. . . . . 6 |- {A} = {A, A}
2		preq2 2446	. . . . . 6 |- ({A} = {A, A} -> {{A}, {A}} = {{A}, {A, A}})
3	1, 2	ax-mp 7	. . . . 5 |- {{A}, {A}} = {{A}, {A, A}}
4		dfsn2 2417	. . . . 5 |- {{A}} = {{A}, {A}}
5		df-op 2413	. . . . 5 |- <.A, A>. = {{A}, {A, A}}
6	3, 4, 5	3eqtr4r 1504	. . . 4 |- <.A, A>. = {{A}}
7	6	sneqi 2415	. . 3 |- {<.A, A>.} = {{{A}}}
8	7	dmeqi 3308	. 2 |- dom {<.A, A>.} = dom {{{A}}}
9		dmsnop 3324	. 2 |- dom {<.A, A>.} = {A}
10	8, 9	eqtr3 1495	1 |- dom {{{A}}} = {A}

Syntax hints:   = wceq 955  {csn 2406  {cpr 2407  <.cop 2408  dom cdm 3166

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-dm 3184

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