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Mirrors > Home > MPE Home > Th. List > iunssd | Structured version Visualization version GIF version |
Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
iunssd.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
iunssd | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunssd.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) | |
2 | 1 | ralrimiva 3133 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
3 | iunss 4835 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | |
4 | 2, 3 | sylibr 226 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2050 ∀wral 3089 ⊆ wss 3830 ∪ ciun 4792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-in 3837 df-ss 3844 df-iun 4794 |
This theorem is referenced by: imasaddfnlem 16657 imasaddflem 16659 subdrgint 19304 meaiininclem 42197 smflim 42482 smfresal 42492 smfmullem4 42498 |
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