Theorem eliund 4961

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 4958	. 2 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
3	1, 2	sylibr 237	1 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∃wrex 3088  ∪ ciun 4954

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-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3455  df-iun 4956

This theorem is used by:  bdayiun  28188  gsumwrd2dccatlem  33525  hoicvr  47384  imaid  50088