![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > untint | Structured version Visualization version GIF version |
Description: If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.) |
Ref | Expression |
---|---|
untint | ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intss1 4968 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥) | |
2 | ssralv 4064 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦)) |
4 | 3 | rexlimiv 3146 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ∩ cint 4951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-v 3480 df-ss 3980 df-int 4952 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |