Theorem elsncg 2482

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

Assertion

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

Proof of Theorem elsncg

Step	Hyp	Ref	Expression
1		elprg 2475	. 2 |- (A e. C -> (A e. {B, B} <-> (A = B \/ A = B)))
2		dfsn2 2472	. . . 4 |- {B} = {B, B}
3	2	eqcomi 1526	. . 3 |- {B, B} = {B}
4	3	eleq2i 1585	. 2 |- (A e. {B, B} <-> A e. {B})
5		oridm 250	. 2 |- ((A = B \/ A = B) <-> A = B)
6	1, 4, 5	3bitr3g 565	1 |- (A e. C -> (A e. {B} <-> A = B))

Syntax hints:   -> wi 3   <-> wb 153   \/ wo 229   = wceq 997   e. wcel 999  {csn 2461  {cpr 2462

This theorem is referenced by:  elsnc 2483  elsni 2484  snidg 2485  eldifsn 2516  elsucg 3093  ltxr 5560

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-v 1859  df-un 2101  df-sn 2464  df-pr 2465

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