Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > iniin2 | Structured version Visualization version GIF version |
Description: Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
iniin2 | ⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iinin2 5002 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶)) | |
2 | 1 | eqcomd 2829 | 1 ⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ≠ wne 3018 ∩ cin 3937 ∅c0 4293 ∩ ciin 4922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-v 3498 df-dif 3941 df-in 3945 df-nul 4294 df-iin 4924 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |