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Mirrors > Home > MPE Home > Th. List > ss2in | Structured version Visualization version GIF version |
Description: Intersection of subclasses. (Contributed by NM, 5-May-2000.) |
Ref | Expression |
---|---|
ss2in | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrin 4207 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) | |
2 | sslin 4208 | . 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐵 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷)) | |
3 | 1, 2 | sylan9ss 3977 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∩ cin 3932 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 |
This theorem is referenced by: disjxiun 5054 undom 8593 strleun 16579 dprdss 19080 dprd2da 19093 ablfac1b 19121 tgcl 21505 innei 21661 hausnei2 21889 bwth 21946 fbssfi 22373 fbunfip 22405 fgcl 22414 blin2 22966 vtxdun 27190 vtxdginducedm1 27252 5oai 29365 mayetes3i 29433 mdsl0 30014 neibastop1 33604 ismblfin 34814 heibor1lem 34968 pl42lem2N 36996 pl42lem3N 36997 ntrk2imkb 40265 ssin0 41194 |
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