Theorem tsksuc 10784

Description: If an element of a Tarski class is an ordinal number, its successor is an element of the class. JFM CLASSES2 th. 6 (partly). (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
tsksuc	⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → suc 𝐴 ∈ 𝑇)

Proof of Theorem tsksuc

Step	Hyp	Ref	Expression
1		simp1 1136	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ Tarski)
2		tskpw 10775	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
3	2	3adant2 1131	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
4		eloni 6373	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
5	4	3ad2ant2 1134	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → Ord 𝐴)
6		ordunisuc 7834	. . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
7		eqimss 4022	. . . 4 ⊢ (∪ suc 𝐴 = 𝐴 → ∪ suc 𝐴 ⊆ 𝐴)
8	5, 6, 7	3syl 18	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → ∪ suc 𝐴 ⊆ 𝐴)
9		sspwuni 5080	. . 3 ⊢ (suc 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ suc 𝐴 ⊆ 𝐴)
10	8, 9	sylibr 234	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → suc 𝐴 ⊆ 𝒫 𝐴)
11		tskss 10780	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇 ∧ suc 𝐴 ⊆ 𝒫 𝐴) → suc 𝐴 ∈ 𝑇)
12	1, 3, 10, 11	syl3anc 1372	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → suc 𝐴 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ⊆ wss 3931  𝒫 cpw 4580  ∪ cuni 4887  Ord word 6362  Oncon0 6363  suc csuc 6365  Tarskictsk 10770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-tr 5240  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-ord 6366  df-on 6367  df-suc 6369  df-tsk 10771

This theorem is referenced by:  tsk2  10787