Theorem inv1 4348

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 4205	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3989	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3991	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 4208	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3983	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1533  Vcvv 3495   ∩ cin 3935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-in 3943  df-ss 3952

This theorem is referenced by:  undif1  4424  dfif4  4482  rint0  4909  iinrab2  4985  riin0  4997  xpriindi  5702  xpssres  5884  resdmdfsn  5896  elrid  5908  imainrect  6033  xpima  6034  cnvrescnv  6047  dmresv  6052  curry1  7793  curry2  7796  fpar  7805  oev2  8142  hashresfn  13694  dmhashres  13695  gsumxp  19090  pjpm  20846  ptbasfi  22183  mbfmcst  31512  0rrv  31704  pol0N  37039