Theorem elsncg 2430

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 2423	. 2 |- (A e. C -> (A e. {B, B} <-> (A = B \/ A = B)))
2		dfsn2 2420	. . . 4 |- {B} = {B, B}
3	2	eqcomi 1479	. . 3 |- {B, B} = {B}
4	3	eleq2i 1538	. 2 |- (A e. {B, B} <-> A e. {B})
5		oridm 243	. 2 |- ((A = B \/ A = B) <-> A = B)
6	1, 4, 5	3bitr3g 554	1 |- (A e. C -> (A e. {B} <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   = wceq 956   e. wcel 958  {csn 2409  {cpr 2410

This theorem is referenced by:  elsnc 2431  elsni 2432  snidg 2433  eldifsn 2462  elsucg 3036  ltxrt 5495

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413

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