Theorem inv1 4394

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

Step	Hyp	Ref	Expression
1		inss1 4228	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 4004	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 4006	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 4231	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3998	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1540  Vcvv 3473   ∩ cin 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965

This theorem is referenced by:  undif1  4475  dfif4  4543  rint0  4994  iinrab2  5073  riin0  5085  xpriindi  5836  xpssres  6018  resdmdfsn  6031  elrid  6045  imainrect  6180  xpima  6181  cnvrescnv  6194  dmresv  6199  curry1  8094  curry2  8097  fpar  8106  oev2  8527  hashresfn  14305  dmhashres  14306  gsumxp  19886  pjpm  21483  ptbasfi  23306  mbfmcst  33557  0rrv  33749  pol0N  39084