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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 2970 | . 2 ⊢ Ⅎ𝑥𝐶 | |
2 | 1 | ssiinf 4790 | 1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∀wral 3118 ⊆ wss 3799 ∩ ciin 4742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-v 3417 df-in 3806 df-ss 3813 df-iin 4744 |
This theorem is referenced by: triin 4991 cflim2 9401 ptbasfi 21756 limciun 24058 clsint2 32863 fnemeet2 32901 dihglblem4 37373 dihglblem6 37416 iooiinicc 40565 iooiinioc 40579 iinhoiicc 41683 smfsuplem1 41812 |
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