Theorem tskssel 10795

Description: A part of a Tarski class strictly dominated by the class is an element of the class. JFM CLASSES2 th. 2. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
tskssel	⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)

Proof of Theorem tskssel

Step	Hyp	Ref	Expression
1		sdomnen 9020	. . 3 ⊢ (𝐴 ≺ 𝑇 → ¬ 𝐴 ≈ 𝑇)
2	1	3ad2ant3 1134	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → ¬ 𝐴 ≈ 𝑇)
3		tsken 10792	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
4	3	3adant3 1131	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
5	4	ord 864	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → (¬ 𝐴 ≈ 𝑇 → 𝐴 ∈ 𝑇))
6	2, 5	mpd 15	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 847   ∧ w3a 1086   ∈ wcel 2106   ⊆ wss 3963   class class class wbr 5148   ≈ cen 8981   ≺ csdm 8983  Tarskictsk 10786

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  ax-sep 5302

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-sdom 8987  df-tsk 10787

This theorem is referenced by:  tskpr  10808  tskwe2  10811  tskord  10818  tskcard  10819  tskurn  10827