Theorem snelpwg 14415

Description: A singleton of a set belongs to the power class of a class containing the set.

Assertion

Ref	Expression
snelpwg	|- (A e. C -> (A e. B <-> {A} e. ~PB))

Proof of Theorem snelpwg

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (A e. C -> A e. _V)
2		eleq1 1957	. . . 4 |- (A = if(A e. _V, A, (/)) -> (A e. B <-> if(A e. _V, A, (/)) e. B))
3		sneq 3054	. . . . 5 |- (A = if(A e. _V, A, (/)) -> {A} = {if(A e. _V, A, (/))})
4	3	eleq1d 1963	. . . 4 |- (A = if(A e. _V, A, (/)) -> ({A} e. ~PB <-> {if(A e. _V, A, (/))} e. ~PB))
5	2, 4	bibi12d 691	. . 3 |- (A = if(A e. _V, A, (/)) -> ((A e. B <-> {A} e. ~PB) <-> (if(A e. _V, A, (/)) e. B <-> {if(A e. _V, A, (/))} e. ~PB)))
6		0ex 3446	. . . . 5 |- (/) e. _V
7	6	elimel 3025	. . . 4 |- if(A e. _V, A, (/)) e. _V
8	7	snelpw 3501	. . 3 |- (if(A e. _V, A, (/)) e. B <-> {if(A e. _V, A, (/))} e. ~PB)
9	5, 8	dedth 3011	. 2 |- (A e. _V -> (A e. B <-> {A} e. ~PB))
10	1, 9	syl 12	1 |- (A e. C -> (A e. B <-> {A} e. ~PB))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  ifcif 2982  ~Pcpw 3032  {csn 3044

This theorem is referenced by:  prsubrtr 14763

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049

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