ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inv1 Structured version   GIF version

Theorem inv1 3247
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
StepHypRef Expression
1 inss1 3151 . 2 (A ∩ V) ⊆ A
2 ssid 2958 . . 3 AA
3 ssv 2959 . . 3 A ⊆ V
42, 3ssini 3154 . 2 A ⊆ (A ∩ V)
51, 4eqssi 2955 1 (A ∩ V) = A
Colors of variables: wff set class
Syntax hints:   = wceq 1242  Vcvv 2551  cin 2910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925
This theorem is referenced by:  rint0  3645  riin0  3719  xpssres  4588  imainrect  4709  xpima2m  4711  dmresv  4722
  Copyright terms: Public domain W3C validator