Theorem eliund 4964

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 4961	. 2 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
3	1, 2	sylibr 237	1 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2143  ∃wrex 3089  ∪ ciun 4957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-v 3457  df-iun 4959

This theorem is referenced by:  bdayiun  28086  gsumwrd2dccatlem  33375  hoicvr  47242  imaid  49909