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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 2955 | . 2 ⊢ Ⅎ𝑥𝐶 | |
2 | 1 | ssiinf 4941 | 1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wral 3106 ⊆ 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: triin 5151 cflim2 9674 ptbasfi 22186 limciun 24497 clsint2 33790 fnemeet2 33828 dihglblem4 38593 dihglblem6 38636 iooiinicc 42179 iooiinioc 42193 iinhoiicc 43313 smfsuplem1 43442 |
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