Theorem inv1 4347

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 4204	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3988	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3990	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 4207	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3982	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1533  Vcvv 3494   ∩ cin 3934

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 2157  ax-12 2173  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 3496  df-in 3942  df-ss 3951

This theorem is referenced by:  undif1  4423  dfif4  4481  rint0  4908  iinrab2  4984  riin0  4996  xpriindi  5701  xpssres  5883  resdmdfsn  5895  elrid  5907  imainrect  6032  xpima  6033  cnvrescnv  6046  dmresv  6051  curry1  7793  curry2  7796  fpar  7805  oev2  8142  hashresfn  13694  dmhashres  13695  gsumxp  19090  pjpm  20846  ptbasfi  22183  mbfmcst  31512  0rrv  31704  pol0N  37039