![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > trintss | Structured version Visualization version GIF version |
Description: Any nonempty transitive class includes its intersection. Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the nonemptiness hypothesis). (Contributed by Scott Fenton, 3-Mar-2011.) (Proof shortened by Andrew Salmon, 14-Nov-2011.) |
Ref | Expression |
---|---|
trintss | ⊢ ((Tr 𝐴 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4339 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | intss1 4958 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥) | |
3 | trss 5267 | . . . . . 6 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
4 | 3 | com12 32 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → 𝑥 ⊆ 𝐴)) |
5 | sstr2 3982 | . . . . 5 ⊢ (∩ 𝐴 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → ∩ 𝐴 ⊆ 𝐴)) | |
6 | 2, 4, 5 | sylsyld 61 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴)) |
7 | 6 | exlimiv 1925 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴)) |
8 | 1, 7 | sylbi 216 | . 2 ⊢ (𝐴 ≠ ∅ → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴)) |
9 | 8 | impcom 407 | 1 ⊢ ((Tr 𝐴 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 ⊆ wss 3941 ∅c0 4315 ∩ cint 4941 Tr wtr 5256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-v 3468 df-dif 3944 df-in 3948 df-ss 3958 df-nul 4316 df-uni 4901 df-int 4942 df-tr 5257 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |