Theorem elsncgOLD 3064

Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized).

Assertion

Ref	Expression
elsncgOLD	|- (A e. C -> (A e. {B} <-> A = B))

Proof of Theorem elsncgOLD

Step	Hyp	Ref	Expression
1		elprg 3060	. 2 |- (A e. C -> (A e. {B, B} <-> (A = B \/ A = B)))
2		dfsn2 3057	. . . 4 |- {B} = {B, B}
3	2	eqcomi 1888	. . 3 |- {B, B} = {B}
4	3	eleq2i 1961	. 2 |- (A e. {B, B} <-> A e. {B})
5		oridm 262	. 2 |- ((A = B \/ A = B) <-> A = B)
6	1, 4, 5	3bitr3g 613	1 |- (A e. C -> (A e. {B} <-> A = B))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   = wceq 1298   e. wcel 1300  {csn 3044  {cpr 3045

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-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

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-v 2294  df-un 2600  df-sn 3049  df-pr 3050

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