Theorem tskssel 9579

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 7984	. . 3 ⊢ (𝐴 ≺ 𝑇 → ¬ 𝐴 ≈ 𝑇)
2	1	3ad2ant3 1084	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → ¬ 𝐴 ≈ 𝑇)
3		tsken 9576	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
4	3	3adant3 1081	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
5	4	ord 392	. 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 383   ∧ w3a 1037   ∈ wcel 1990   ⊆ wss 3574   class class class wbr 4653   ≈ cen 7952   ≺ csdm 7954  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-sdom 7958  df-tsk 9571

This theorem is referenced by:  tskpr  9592  tskwe2  9595  tskord  9602  tskcard  9603  tskurn  9611