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Mirrors > Home > MPE Home > Th. List > ssiin | Structured version Visualization version GIF version |
Description: Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.) |
Ref | Expression |
---|---|
ssiin | ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2793 | . 2 ⊢ Ⅎ𝑥𝐶 | |
2 | 1 | ssiinf 4601 | 1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∀wral 2941 ⊆ wss 3607 ∩ ciin 4553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-v 3233 df-in 3614 df-ss 3621 df-iin 4555 |
This theorem is referenced by: cflim2 9123 ptbasfi 21432 limciun 23703 clsint2 32449 fnemeet2 32487 dihglblem4 36903 dihglblem6 36946 iooiinicc 40087 iooiinioc 40101 iinhoiicc 41209 smfsuplem1 41338 |
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