MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intsn Structured version   Visualization version   GIF version

Theorem intsn 4665
Description: The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)
Hypothesis
Ref Expression
intsn.1 𝐴 ∈ V
Assertion
Ref Expression
intsn {𝐴} = 𝐴

Proof of Theorem intsn
StepHypRef Expression
1 intsn.1 . 2 𝐴 ∈ V
2 intsng 4664 . 2 (𝐴 ∈ V → {𝐴} = 𝐴)
31, 2ax-mp 5 1 {𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321   cint 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-un 3720  df-in 3722  df-sn 4322  df-pr 4324  df-int 4628
This theorem is referenced by:  uniintsn  4666  intunsn  4668  op1stb  5088  op2ndb  5780  ssfii  8492  cf0  9285  cflecard  9287  uffix  21946  iotain  39138
  Copyright terms: Public domain W3C validator