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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 2684 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elint 3772 | . . 3 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)) |
3 | eleq1 2200 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
4 | eleq2 2201 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴)) | |
5 | 3, 4 | imbi12d 233 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) ↔ (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐴))) |
6 | 5 | spcgv 2768 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐴))) |
7 | 6 | pm2.43a 51 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝐴)) |
8 | 2, 7 | syl5bi 151 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ 𝐴)) |
9 | 8 | ssrdv 3098 | 1 ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 = wceq 1331 ∈ wcel 1480 ⊆ wss 3066 ∩ cint 3766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-int 3767 |
This theorem is referenced by: intminss 3791 intmin3 3793 intab 3795 int0el 3796 trintssm 4037 inteximm 4069 onnmin 4478 peano5 4507 peano5nnnn 7693 peano5nni 8716 dfuzi 9154 bj-intabssel 12985 bj-intabssel1 12986 |
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