Theorem inv1 4351

Description: The intersection of a class with the universal class is itself. Dual of un0 4347. 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 4136	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3935	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3937	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 4139	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3929	1 ⊢ (𝐴 ∩ V) = 𝐴

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   ∩ cin 4092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-in 4094

This theorem is used by:  vvin  4362  undif1  4433  dfif4  4501  rint0  4953  iinrab2  5033  riin0  5047  xpriindi  5823  xpres  6024  resdmdfsn  6039  resdmdfsnOLD  6040  elrid  6056  imainrect  6191  xpima  6192  cnvrescnv  6206  dmresv  6213  imadifssranOLD  6221  curry1  8152  curry2  8155  fpar  8164  oev2  8569  hashresfn  14525  dmhashres  14526  gsumxp  20241  pjpm  22069  ptbasfi  23955  inv2  35822  fineqvomon  35927  vonf1wev  36077  vonf1owevOLD  36079  ecqmap  39647  pol0N  41232