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Theorem inv1 3577
 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 3475 . 2 (A ∩ V) A
2 ssid 3290 . . 3 A A
3 ssv 3291 . . 3 A V
42, 3ssini 3478 . 2 A (A ∩ V)
51, 4eqssi 3288 1 (A ∩ V) = A
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  Vcvv 2859   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  undif1  3625  dfif4  3673  rint0  3966  iinrab2  4029  riin0  4039  compldif  4069  xpkexg  4288
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