Theorem tskssel 10715

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 8962	. . 3 ⊢ (𝐴 ≺ 𝑇 → ¬ 𝐴 ≈ 𝑇)
2	1	3ad2ant3 1148	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → ¬ 𝐴 ≈ 𝑇)
3		tsken 10712	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
4	3	3adant3 1145	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
5	4	ord 875	. 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 858   ∧ w3a 1098   ∈ wcel 2142   ⊆ wss 3904   class class class wbr 5100   ≈ cen 8924   ≺ csdm 8926  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-sdom 8930  df-tsk 10707

This theorem is referenced by:  tskpr  10728  tskwe2  10731  tskord  10738  tskcard  10739  tskurn  10747