Theorem elint2 3725

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 3724	. 2 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))
3		df-ral 2380	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))
4	2, 3	bitr4i 186	1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1297   ∈ wcel 1448  ∀wral 2375  Vcvv 2641  ∩ cint 3718

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-int 3719

This theorem is referenced by:  elintg  3726  ssint  3734  intssunim  3740  iinuniss  3841  trint  3981