Theorem elint 3928

Description: Membership in class intersection. (Contributed by NM, 21-May-1994.)

Hypothesis

Ref	Expression
elint.1	⊢ 𝐴 ∈ V

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elint

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elint.1	. 2 ⊢ 𝐴 ∈ V
2		eleq1 2292	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
3	2	imbi2d 230	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)))
4	3	albidv 1870	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)))
5		df-int 3923	. 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥)}
6	1, 4, 5	elab2 2951	1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1393   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ∩ cint 3922

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-int 3923

This theorem is referenced by:  elint2  3929  elintab  3933  intss1  3937  intss  3943  intun  3953  intpr  3954  peano1  4683