Theorem elsn 3749

Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
elsn	⊢ (x ∈ {A} ↔ x = A)

Distinct variable group:   x,A

Proof of Theorem elsn

Step	Hyp	Ref	Expression
1		df-sn 3742	. 2 ⊢ {A} = {x ∣ x = A}
2	1	abeq2i 2461	1 ⊢ (x ∈ {A} ↔ x = A)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sn 3742

This theorem is referenced by:  dfpr2  3750  ralsns  3764  rexsns  3765  disjsn  3787  snprc  3789  euabsn2  3792  snss  3839  difprsnss  3847  pwpw0  3856  eqsn  3868  snsspw  3878  pwsnALT  3883  dfnfc2  3910  uni0b  3917  uni0c  3918  axprimlem1  4089  pwadjoin  4120  pw10  4162  pw1sn  4166  eqpw1uni  4331  nnsucelrlem2  4426  opeliunxp  4821  dmsnopg  5067  nfunsn  5354  fsn  5433  fconstfv  5457  foundex  5915  el2c  6192  csucex  6260  nmembers1lem3  6271  nchoicelem6  6295