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Mirrors > Home > ILE Home > Th. List > intss1 | GIF version |
Description: An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.) |
Ref | Expression |
---|---|
intss1 | ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elint 3621 | . . 3 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)) |
3 | eleq1 2100 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
4 | eleq2 2101 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴)) | |
5 | 3, 4 | imbi12d 223 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) ↔ (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐴))) |
6 | 5 | spcgv 2640 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐴))) |
7 | 6 | pm2.43a 45 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝐴)) |
8 | 2, 7 | syl5bi 141 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ 𝐴)) |
9 | 8 | ssrdv 2951 | 1 ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 = wceq 1243 ∈ wcel 1393 ⊆ wss 2917 ∩ cint 3615 |
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-v 2559 df-in 2924 df-ss 2931 df-int 3616 |
This theorem is referenced by: intminss 3640 intmin3 3642 intab 3644 int0el 3645 trint0m 3871 inteximm 3903 onnmin 4292 peano5 4321 peano5nnnn 6966 peano5nni 7917 dfuzi 8348 bj-intabssel 9928 bj-intabssel1 9929 peano5setOLD 10065 |
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