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Mirrors > Home > MPE Home > Th. List > tskin | Structured version Visualization version GIF version |
Description: The intersection of two elements of a Tarski class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
tskin | ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐴 ∩ 𝐵) ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 3833 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | tskss 9580 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐴 ∩ 𝐵) ∈ 𝑇) | |
3 | 1, 2 | mp3an3 1413 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐴 ∩ 𝐵) ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 Tarskictsk 9570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-tsk 9571 |
This theorem is referenced by: (None) |
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