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Mirrors > Home > ILE Home > Th. List > trint0m | GIF version |
Description: Any inhabited transitive class includes its intersection. Similar to Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Jim Kingdon, 22-Aug-2018.) |
Ref | Expression |
---|---|
trint0m | ⊢ ((Tr A ∧ ∃x x ∈ A) → ∩ A ⊆ A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intss1 3621 | . . . 4 ⊢ (x ∈ A → ∩ A ⊆ x) | |
2 | trss 3854 | . . . . 5 ⊢ (Tr A → (x ∈ A → x ⊆ A)) | |
3 | 2 | com12 27 | . . . 4 ⊢ (x ∈ A → (Tr A → x ⊆ A)) |
4 | sstr2 2946 | . . . 4 ⊢ (∩ A ⊆ x → (x ⊆ A → ∩ A ⊆ A)) | |
5 | 1, 3, 4 | sylsyld 52 | . . 3 ⊢ (x ∈ A → (Tr A → ∩ A ⊆ A)) |
6 | 5 | exlimiv 1486 | . 2 ⊢ (∃x x ∈ A → (Tr A → ∩ A ⊆ A)) |
7 | 6 | impcom 116 | 1 ⊢ ((Tr A ∧ ∃x x ∈ A) → ∩ A ⊆ A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∃wex 1378 ∈ wcel 1390 ⊆ wss 2911 ∩ cint 3606 Tr wtr 3845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-in 2918 df-ss 2925 df-uni 3572 df-int 3607 df-tr 3846 |
This theorem is referenced by: (None) |
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