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Theorem elint2 3866
Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
Hypothesis
Ref Expression
elint2.1 𝐴 ∈ V
Assertion
Ref Expression
elint2 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elint2
StepHypRef Expression
1 elint2.1 . . 3 𝐴 ∈ V
21elint 3865 . 2 (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
3 df-ral 2473 . 2 (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
42, 3bitr4i 187 1 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1362  wcel 2160  wral 2468  Vcvv 2752   cint 3859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-int 3860
This theorem is referenced by:  elintg  3867  ssint  3875  intssunim  3881  iinuniss  3984  trint  4131  suplocexprlemmu  7747  suplocexprlemdisj  7749  suplocexprlemloc  7750  suplocexprlemub  7752
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