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Theorem inv1 3399
 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 3296 . 2 (𝐴 ∩ V) ⊆ 𝐴
2 ssid 3117 . . 3 𝐴𝐴
3 ssv 3119 . . 3 𝐴 ⊆ V
42, 3ssini 3299 . 2 𝐴 ⊆ (𝐴 ∩ V)
51, 4eqssi 3113 1 (𝐴 ∩ V) = 𝐴
 Colors of variables: wff set class Syntax hints:   = wceq 1331  Vcvv 2686   ∩ cin 3070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084 This theorem is referenced by:  rint0  3810  riin0  3884  xpssres  4854  resdmdfsn  4862  imainrect  4984  xpima2m  4986  dmresv  4997
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