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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 2902 | . 2 ⊢ Ⅎ𝑥𝐶 | |
2 | 1 | ssiinf 5057 | 1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wral 3060 ⊆ wss 3948 ∩ ciin 4998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-v 3475 df-in 3955 df-ss 3965 df-iin 5000 |
This theorem is referenced by: triin 5282 cflim2 10264 ptbasfi 23318 limciun 25656 clsint2 35530 fnemeet2 35568 dihglblem4 40484 dihglblem6 40527 iooiinicc 44566 iooiinioc 44580 iinhoiicc 45701 smfsuplem1 45838 |
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