Theorem eliunid 45980

Description: Membership in indexed union. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Assertion

Ref	Expression
eliunid	⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem eliunid

Step	Hyp	Ref	Expression
1		rspe 3252	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
2		eliun 4955	. 2 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
3	1, 2	sylibr 237	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∃wrex 3086  ∪ ciun 4951

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-rex 3087  df-v 3452  df-iun 4953

This theorem is used by: (None)