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Theorem inv1 4003
 Description: The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
inv1 (𝐴 ∩ V) = 𝐴

Proof of Theorem inv1
StepHypRef Expression
1 inss1 3866 . 2 (𝐴 ∩ V) ⊆ 𝐴
2 ssid 3657 . . 3 𝐴𝐴
3 ssv 3658 . . 3 𝐴 ⊆ V
42, 3ssini 3869 . 2 𝐴 ⊆ (𝐴 ∩ V)
51, 4eqssi 3652 1 (𝐴 ∩ V) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1523  Vcvv 3231   ∩ cin 3606 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621 This theorem is referenced by:  undif1  4076  dfif4  4134  rint0  4549  iinrab2  4615  riin0  4626  xpriindi  5291  xpssres  5469  resdmdfsn  5480  imainrect  5610  xpima  5611  dmresv  5628  curry1  7314  curry2  7317  fpar  7326  oev2  7648  hashresfn  13168  dmhashres  13169  gsumxp  18421  pjpm  20100  ptbasfi  21432  mbfmcst  30449  0rrv  30641  pol0N  35513
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