Theorem elint2 4926

Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.)

Hypothesis

Ref	Expression
elint2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elint2	⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elint2

Step	Hyp	Ref	Expression
1		elint2.1	. . 3 ⊢ 𝐴 ∈ V
2	1	elint 4925	. 2 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))
3		df-ral 3051	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))
4	2, 3	bitr4i 278	1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537   ∈ wcel 2107  ∀wral 3050  Vcvv 3457  ∩ cint 4919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-int 4920

This theorem is referenced by:  int0  4935  ssint  4937  intssuni  4943  iinuni  5071  onint  7778  intwun  10741  inttsk  10780  intgru  10820  subgint  19118  subrngint  20505  subrgint  20540  lssintcl  20906  toponmre  23016  alexsubALTlem3  23972  shintcli  31242  chintcli  31244  intlidl  33353  fin2so  37552  intidl  37974  mzpincl  42682  elimaint  43598  elintima  43602  intsal  46289  salgencntex  46302