Theorem eliinid 45943

Description: Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Assertion

Ref	Expression
eliinid	⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem eliinid

Step	Hyp	Ref	Expression
1		simpl 488	. . 3 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
2		eliin 4956	. . . 4 ⊢ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3	2	adantr 486	. . 3 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
4	1, 3	mpbid 235	. 2 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
5		rspa 3251	. 2 ⊢ ((∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)
6	4, 5	sylancom 600	1 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  ∀wral 3076  ∩ ciin 4952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-iin 4954

This theorem is used by:  iinssiin  45961  fnlimfvre  46502  smflimlem2  47600  smflimmpt  47638  smfsuplem1  47639  smfsupmpt  47643  smfsupxr  47644  smfinflem  47645  smfinfmpt  47647  smflimsuplem4  47651  smflimsupmpt  47657  smfliminfmpt  47660  fsupdm  47670  finfdm  47674