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| 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.) | 
| Ref | Expression | 
|---|---|
| tsksuc | ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → suc 𝐴 ∈ 𝑇) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simp1 1136 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ Tarski) | |
| 2 | tskpw 10794 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇) | |
| 3 | 2 | 3adant2 1131 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇) | 
| 4 | eloni 6393 | . . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 5 | 4 | 3ad2ant2 1134 | . . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → Ord 𝐴) | 
| 6 | ordunisuc 7853 | . . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
| 7 | eqimss 4041 | . . . 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 5099 | . . 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 10799 | . 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) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 𝒫 cpw 4599 ∪ cuni 4906 Ord word 6382 Oncon0 6383 suc csuc 6385 Tarskictsk 10789 | 
| 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 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| 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 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-suc 6389 df-tsk 10790 | 
| This theorem is referenced by: tsk2 10806 | 
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