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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 4796 | . . 3 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)) | |
2 | eleq2 2873 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐶)) | |
3 | 2 | rspccv 3558 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)) |
4 | 1, 3 | syl6bi 254 | . 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 2083 ∀wral 3107 ∩ cint 4788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-v 3442 df-int 4789 |
This theorem is referenced by: inttsk 10049 subgint 18061 subrgint 19251 lssintcl 19430 ufinffr 22225 shintcli 28793 insiga 31009 intsal 42177 |
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