Theorem eliunid 45148

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 3228	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
2		eliun 4962	. 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 2109  ∃wrex 3054  ∪ ciun 4958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rex 3055  df-v 3452  df-iun 4960

This theorem is referenced by: (None)