Theorem elint2 4958

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 4957	. 2 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))
3		df-ral 3063	. 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 205  ∀wal 1540   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-int 4952

This theorem is referenced by:  int0  4967  ssint  4969  intssuni  4975  iinuni  5102  onint  7778  intwun  10730  inttsk  10769  intgru  10809  subgint  19030  subrgint  20342  lssintcl  20575  toponmre  22597  alexsubALTlem3  23553  shintcli  30613  chintcli  30615  intlidl  32567  fin2so  36523  intidl  36945  mzpincl  41520  elimaint  42448  elintima  42452  intsal  45094  salgencntex  45107  subrngint  46787