Theorem elintg 3882

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 2259	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ∩ 𝐵 ↔ 𝐴 ∈ ∩ 𝐵))
2		eleq1 2259	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
3	2	ralbidv 2497	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥))
4		vex 2766	. . 3 ⊢ 𝑦 ∈ V
5	4	elint2 3881	. 2 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
6	1, 3, 5	vtoclbg 2825	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∩ cint 3874

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-int 3875

This theorem is referenced by:  elinti  3883  elrint  3914  peano2  4631  pitonn  7915  peano1nnnn  7919  peano2nnnn  7920  1nn  9001  peano2nn  9002  subgintm  13328  subrngintm  13768  subrgintm  13799  lssintclm  13940