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Theorem inv1 3338
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 3235 . 2 (𝐴 ∩ V) ⊆ 𝐴
2 ssid 3059 . . 3 𝐴𝐴
3 ssv 3061 . . 3 𝐴 ⊆ V
42, 3ssini 3238 . 2 𝐴 ⊆ (𝐴 ∩ V)
51, 4eqssi 3055 1 (𝐴 ∩ V) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1296  Vcvv 2633  cin 3012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026
This theorem is referenced by:  rint0  3749  riin0  3823  xpssres  4780  resdmdfsn  4788  imainrect  4910  xpima2m  4912  dmresv  4923
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