Theorem elint2 4913

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 4912	. 2 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))
3		df-ral 3045	. 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 1538   ∈ wcel 2109  ∀wral 3044  Vcvv 3444  ∩ cint 4906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-int 4907

This theorem is referenced by:  int0  4922  ssint  4924  intssuni  4930  iinuni  5057  onint  7746  intwun  10664  inttsk  10703  intgru  10743  subgint  19058  subrngint  20445  subrgint  20480  lssintcl  20846  toponmre  22956  alexsubALTlem3  23912  shintcli  31231  chintcli  31233  intlidl  33364  fin2so  37574  intidl  37996  mzpincl  42695  elimaint  43611  elintima  43615  intsal  46301  salgencntex  46314