![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > eliinid | Structured version Visualization version GIF version |
Description: Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
eliinid | ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . 3 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶) | |
2 | eliin 4958 | . . . 4 ⊢ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) | |
3 | 2 | adantr 482 | . . 3 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
4 | 1, 3 | mpbid 231 | . 2 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
5 | rspa 3230 | . 2 ⊢ ((∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶) | |
6 | 4, 5 | sylancom 589 | 1 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∀wral 3063 ∩ ciin 4954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-iin 4956 |
This theorem is referenced by: iinssiin 43244 fnlimfvre 43810 smflimlem2 44908 smflimmpt 44946 smfsuplem1 44947 smfsupmpt 44951 smfsupxr 44952 smfinflem 44953 smfinfmpt 44955 smflimsuplem4 44959 smflimsupmpt 44965 smfliminfmpt 44968 fsupdm 44978 finfdm 44982 |
Copyright terms: Public domain | W3C validator |