Theorem elintg 3779

Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elintg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2202	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ∩ 𝐵 ↔ 𝐴 ∈ ∩ 𝐵))
2		eleq1 2202	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
3	2	ralbidv 2437	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥))
4		vex 2689	. . 3 ⊢ 𝑦 ∈ V
5	4	elint2 3778	. 2 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
6	1, 3, 5	vtoclbg 2747	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∩ cint 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-int 3772

This theorem is referenced by:  elinti  3780  elrint  3811  peano2  4509  pitonn  7656  peano1nnnn  7660  peano2nnnn  7661  1nn  8731  peano2nn  8732