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Mirrors > Home > ILE Home > Th. List > trintssm | GIF version |
Description: If 𝐴 is transitive and inhabited, then ∩ 𝐴 is a subset of 𝐴. (Contributed by Jim Kingdon, 22-Aug-2018.) |
Ref | Expression |
---|---|
trintssm | ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ Tr 𝐴) → ∩ 𝐴 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | 1 | elint2 3622 | . . 3 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) |
3 | r19.2m 3309 | . . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
4 | 3 | ex 108 | . . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)) |
5 | trel 3861 | . . . . . 6 ⊢ (Tr 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴)) | |
6 | 5 | expcomd 1330 | . . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))) |
7 | 6 | rexlimdv 2432 | . . . 4 ⊢ (Tr 𝐴 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) |
8 | 4, 7 | sylan9 389 | . . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ Tr 𝐴) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) |
9 | 2, 8 | syl5bi 141 | . 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ Tr 𝐴) → (𝑦 ∈ ∩ 𝐴 → 𝑦 ∈ 𝐴)) |
10 | 9 | ssrdv 2951 | 1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ Tr 𝐴) → ∩ 𝐴 ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∃wex 1381 ∈ wcel 1393 ∀wral 2306 ∃wrex 2307 ⊆ wss 2917 ∩ cint 3615 Tr wtr 3854 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-int 3616 df-tr 3855 |
This theorem is referenced by: (None) |
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