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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliunid | Structured version Visualization version GIF version |
Description: Membership in indexed union. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
eliunid | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspe 3246 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) | |
2 | eliun 4999 | . 2 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) | |
3 | 1, 2 | sylibr 234 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ∃wrex 3067 ∪ ciun 4995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3068 df-v 3479 df-iun 4997 |
This theorem is referenced by: (None) |
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