Theorem snidg 1828

Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49.

Assertion

Ref	Expression
snidg	⊢ (A ∈ B → A ∈ {A})

Proof of Theorem snidg

Step	Hyp	Ref	Expression
1		cleqid 1102	. 2 ⊢ A = A
2		elsncg 1825	. 2 ⊢ (A ∈ B → (A ∈ {A} ↔ A = A))
3	1, 2	mpbiri 169	1 ⊢ (A ∈ B → A ∈ {A})

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  {csn 1808

This theorem is referenced by:  snidb 1829  elsnc2g 1831  disjsn 1836  supsn 2168  cfsuc 3709

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812

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