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Theorem inv1 3281
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 3185 . 2 (𝐴 ∩ V) ⊆ 𝐴
2 ssid 2992 . . 3 𝐴𝐴
3 ssv 2993 . . 3 𝐴 ⊆ V
42, 3ssini 3188 . 2 𝐴 ⊆ (𝐴 ∩ V)
51, 4eqssi 2989 1 (𝐴 ∩ V) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1259  Vcvv 2574  cin 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959
This theorem is referenced by:  rint0  3682  riin0  3756  xpssres  4673  imainrect  4794  xpima2m  4796  dmresv  4807
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