| 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 3249 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) | |
| 2 | eliun 4995 | . 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 2108 ∃wrex 3070 ∪ ciun 4991 |
| 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 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-v 3482 df-iun 4993 |
| This theorem is referenced by: (None) |
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