Theorem inv1 3970

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 3833	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3624	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3625	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3836	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3619	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1483  Vcvv 3200   ∩ cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by:  undif1  4043  dfif4  4101  rint0  4517  iinrab2  4583  riin0  4594  xpriindi  5258  xpssres  5434  resdmdfsn  5445  imainrect  5575  xpima  5576  dmresv  5593  curry1  7269  curry2  7272  fpar  7281  oev2  7603  hashresfn  13128  dmhashres  13129  gsumxp  18375  pjpm  20052  ptbasfi  21384  mbfmcst  30321  0rrv  30513  pol0N  35195