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Theorem elsnc2g 3761
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
elsnc2g (B V → (A {B} ↔ A = B))

Proof of Theorem elsnc2g
StepHypRef Expression
1 elsni 3757 . 2 (A {B} → A = B)
2 snidg 3758 . . 3 (B VB {B})
3 eleq1 2413 . . 3 (A = B → (A {B} ↔ B {B}))
42, 3syl5ibrcom 213 . 2 (B V → (A = BA {B}))
51, 4impbid2 195 1 (B V → (A {B} ↔ A = B))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   = wceq 1642   wcel 1710  {csn 3737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sn 3741
This theorem is referenced by:  elsnc2  3762  fnfreclem2  6318  fnfreclem3  6319
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