MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elint2 Structured version   Visualization version   GIF version

Theorem elint2 4452
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 4451 . 2 (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
3 df-ral 2912 . 2 (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
42, 3bitr4i 267 1 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wcel 1987  wral 2907  Vcvv 3189   cint 4445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3191  df-int 4446
This theorem is referenced by:  elintgOLD  4454  int0  4460  ssint  4463  intssuni  4469  iinuni  4580  trint  4733  trintssOLD  4735  onint  6949  intwun  9509  inttsk  9548  intgru  9588  subgint  17550  subrgint  18734  lssintcl  18896  toponmre  20820  alexsubALTlem3  21776  shintcli  28058  chintcli  28060  fin2so  33063  intidl  33495  mzpincl  36812  elimaint  37456  elintima  37461  intsal  39881  salgencntex  39894
  Copyright terms: Public domain W3C validator