Theorem eliund 4935

Description: Membership in indexed union. (Contributed by Glauco Siliprandi, 15-Feb-2025.)

Hypothesis

Ref	Expression
eliund.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)

Assertion

Ref	Expression
eliund	⊢ (𝜑 → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem eliund

Step	Hyp	Ref	Expression
1		eliund.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
2		eliun 4932	. 2 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
3	1, 2	sylibr 235	1 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2119  ∃wrex 3064  ∪ ciun 4928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rex 3065  df-v 3434  df-iun 4930

This theorem is referenced by:  bdayiun  27932  gsumwrd2dccatlem  33165  hoicvr  46998  imaid  49651