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| Mirrors > Home > MPE Home > Th. List > elinti | Structured version Visualization version GIF version | ||
| Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) | 
| Ref | Expression | 
|---|---|
| elinti | ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elintg 4953 | . . 3 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)) | |
| 2 | eleq2 2829 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐶)) | |
| 3 | 2 | rspccv 3618 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)) | 
| 4 | 1, 3 | biimtrdi 253 | . 2 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶))) | 
| 5 | 4 | pm2.43i 52 | 1 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3060 ∩ cint 4945 | 
| 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-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-int 4946 | 
| This theorem is referenced by: inttsk 10815 subgint 19169 subrngint 20561 subrgint 20596 lssintcl 20963 ufinffr 23938 shintcli 31349 intlidl 33449 insiga 34139 intsal 46350 | 
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