Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2663 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elint 3747 | . . 3 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)) |
3 | eleq1 2180 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
4 | eleq2 2181 | . . . . . 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 2747 | . . . 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 3073 | 1 ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1314 = wceq 1316 ∈ wcel 1465 ⊆ wss 3041 ∩ cint 3741 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-ss 3054 df-int 3742 |
This theorem is referenced by: intminss 3766 intmin3 3768 intab 3770 int0el 3771 trintssm 4012 inteximm 4044 onnmin 4453 peano5 4482 peano5nnnn 7668 peano5nni 8691 dfuzi 9129 bj-intabssel 12923 bj-intabssel1 12924 |
Copyright terms: Public domain | W3C validator |