Theorem inv1 4196

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 4058	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3849	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3851	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 4061	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3844	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1658  Vcvv 3415   ∩ cin 3798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-v 3417  df-in 3806  df-ss 3813

This theorem is referenced by:  undif1  4267  dfif4  4322  rint0  4738  iinrab2  4804  riin0  4815  xpriindi  5492  xpssres  5670  resdmdfsn  5683  elrid  5695  imainrect  5817  xpima  5818  dmresv  5835  curry1  7534  curry2  7537  fpar  7546  oev2  7871  hashresfn  13421  dmhashres  13422  gsumxp  18729  pjpm  20416  ptbasfi  21756  mbfmcst  30867  0rrv  31060  pol0N  35985