Theorem eliunid 45052

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 3255	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
2		eliun 5019	. 2 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
3	1, 2	sylibr 234	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∃wrex 3076  ∪ ciun 5015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-v 3490  df-iun 5017

This theorem is referenced by: (None)