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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 3986 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) | |
2 | sslin 3987 | . 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐵 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷)) | |
3 | 1, 2 | sylan9ss 3765 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∩ cin 3722 ⊆ wss 3723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-v 3353 df-in 3730 df-ss 3737 |
This theorem is referenced by: disjxiun 4783 undom 8204 strlemor1OLD 16177 strleun 16180 dprdss 18636 dprd2da 18649 ablfac1b 18677 tgcl 20994 innei 21150 hausnei2 21378 bwth 21434 fbssfi 21861 fbunfip 21893 fgcl 21902 blin2 22454 vtxdun 26612 vtxdginducedm1 26674 5oai 28860 mayetes3i 28928 mdsl0 29509 neibastop1 32691 ismblfin 33783 heibor1lem 33940 pl42lem2N 35788 pl42lem3N 35789 ntrk2imkb 38861 ssin0 39744 |
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