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Mirrors > Home > MPE Home > Th. List > inss | Structured version Visualization version GIF version |
Description: Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.) |
Ref | Expression |
---|---|
inss | ⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssinss1 4214 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
2 | incom 4178 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
3 | ssinss1 4214 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 ∩ 𝐴) ⊆ 𝐶) | |
4 | 2, 3 | eqsstrid 4015 | . 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
5 | 1, 4 | jaoi 853 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 ∩ cin 3935 ⊆ wss 3936 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 |
This theorem is referenced by: pmatcoe1fsupp 21309 ppttop 21615 inindif 30278 disjorimxrn 35993 iunrelexp0 40067 ntrclsk3 40440 icccncfext 42190 |
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