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Mirrors > Home > MPE Home > Th. List > ssini | Structured version Visualization version GIF version |
Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.) |
Ref | Expression |
---|---|
ssini.1 | ⊢ 𝐴 ⊆ 𝐵 |
ssini.2 | ⊢ 𝐴 ⊆ 𝐶 |
Ref | Expression |
---|---|
ssini | ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssini.1 | . . 3 ⊢ 𝐴 ⊆ 𝐵 | |
2 | ssini.2 | . . 3 ⊢ 𝐴 ⊆ 𝐶 | |
3 | 1, 2 | pm3.2i 474 | . 2 ⊢ (𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) |
4 | ssin 4157 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) | |
5 | 3, 4 | mpbi 233 | 1 ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∩ cin 3880 ⊆ wss 3881 |
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-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 |
This theorem is referenced by: inv1 4302 cnvrescnv 6019 hartogslem1 8990 xptrrel 14331 fbasrn 22489 limciun 24497 hlimcaui 29019 chdmm1i 29260 chm0i 29273 ledii 29319 lejdii 29321 mayetes3i 29512 mdslj2i 30103 mdslmd2i 30113 sumdmdlem2 30202 sigapildsys 31531 ssoninhaus 33909 bj-disj2r 34464 bj-idres 34575 bj-rvecsscvec 34718 icomnfinre 42189 fouriersw 42873 sge0split 43048 |
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