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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 4254 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
2 | incom 4217 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
3 | ssinss1 4254 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 ∩ 𝐴) ⊆ 𝐶) | |
4 | 2, 3 | eqsstrid 4044 | . 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 3962 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 |
This theorem is referenced by: pmatcoe1fsupp 22723 ppttop 23030 disjorimxrn 38730 iunrelexp0 43692 ntrclsk3 44060 icccncfext 45843 |
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