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| 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 4245 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
| 2 | incom 4208 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
| 3 | ssinss1 4245 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 ∩ 𝐴) ⊆ 𝐶) | |
| 4 | 2, 3 | eqsstrid 4021 | . 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | 
| 5 | 1, 4 | jaoi 857 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ wo 847 ∩ cin 3949 ⊆ wss 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-in 3957 df-ss 3967 | 
| This theorem is referenced by: pmatcoe1fsupp 22708 ppttop 23015 disjorimxrn 38750 iunrelexp0 43720 ntrclsk3 44088 icccncfext 45907 | 
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