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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 4886 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
2 | 1 | elv 3446 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
3 | rsp 3170 | . . . 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 | syl5bi 245 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ 𝐵)) |
6 | 5 | ssrdv 3921 | 1 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 ∩ ciin 4882 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 df-iin 4884 |
This theorem is referenced by: dmiin 5789 gruiin 10221 txtube 22245 iooiinicc 42179 iooiinioc 42193 meaiininclem 43125 smfsuplem1 43442 smfsuplem3 43444 smflimsuplem2 43452 |
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