Theorem tsksuc 10656

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 10647	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
3	2	3adant2 1131	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
4		eloni 6317	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
5	4	3ad2ant2 1134	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → Ord 𝐴)
6		ordunisuc 7765	. . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
7		eqimss 3994	. . . 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 5049	. . 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 10652	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇 ∧ suc 𝐴 ⊆ 𝒫 𝐴) → suc 𝐴 ∈ 𝑇)
12	1, 3, 10, 11	syl3anc 1373	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → suc 𝐴 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  Ord word 6306  Oncon0 6307  suc csuc 6309  Tarskictsk 10642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-tr 5200  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6310  df-on 6311  df-suc 6313  df-tsk 10643

This theorem is referenced by:  tsk2  10659