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Theorem inv1 4113
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 3976 . 2 (𝐴 ∩ V) ⊆ 𝐴
2 ssid 3765 . . 3 𝐴𝐴
3 ssv 3766 . . 3 𝐴 ⊆ V
42, 3ssini 3979 . 2 𝐴 ⊆ (𝐴 ∩ V)
51, 4eqssi 3760 1 (𝐴 ∩ V) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  Vcvv 3340  cin 3714
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-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729
This theorem is referenced by:  undif1  4187  dfif4  4245  rint0  4669  iinrab2  4735  riin0  4746  xpriindi  5414  xpssres  5592  resdmdfsn  5603  imainrect  5733  xpima  5734  dmresv  5751  curry1  7437  curry2  7440  fpar  7449  oev2  7772  hashresfn  13322  dmhashres  13323  gsumxp  18575  pjpm  20254  ptbasfi  21586  mbfmcst  30630  0rrv  30822  pol0N  35698
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