Theorem tskin 10717

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.)

Assertion

Ref	Expression
tskin	⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐴 ∩ 𝐵) ∈ 𝑇)

Proof of Theorem tskin

Step	Hyp	Ref	Expression
1		inss1 4188	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		tskss 10716	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐴 ∩ 𝐵) ∈ 𝑇)
3	1, 2	mp3an3 1471	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐴 ∩ 𝐵) ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142   ∩ cin 3903   ⊆ wss 3904  Tarskictsk 10706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-tsk 10707

This theorem is referenced by: (None)