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Mirrors > Home > MPE Home > Th. List > iinss2 | Structured version Visualization version GIF version |
Description: An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.) |
Ref | Expression |
---|---|
iinss2 | ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliin 4995 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
2 | 1 | elv 3479 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
3 | rsp 3243 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵)) | |
4 | 3 | com12 32 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵)) |
5 | 2, 4 | biimtrid 241 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ 𝐵)) |
6 | 5 | ssrdv 3984 | 1 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 ∀wral 3060 Vcvv 3473 ⊆ wss 3944 ∩ ciin 4991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-v 3475 df-in 3951 df-ss 3961 df-iin 4993 |
This theorem is referenced by: dmiin 5944 gruiin 10787 txtube 23073 iooiinicc 44028 iooiinioc 44042 meaiininclem 44975 smfsuplem1 45300 smfsuplem3 45302 smflimsuplem2 45310 |
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