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Mirrors > Home > MPE Home > Th. List > tsksuc | Structured version Visualization version GIF version |
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 1061 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ Tarski) | |
2 | tskpw 9575 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇) | |
3 | 2 | 3adant2 1080 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇) |
4 | eloni 5733 | . . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
5 | 4 | 3ad2ant2 1083 | . . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → Ord 𝐴) |
6 | ordunisuc 7032 | . . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
7 | eqimss 3657 | . . . 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 4611 | . . 3 ⊢ (suc 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ suc 𝐴 ⊆ 𝐴) | |
10 | 8, 9 | sylibr 224 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → suc 𝐴 ⊆ 𝒫 𝐴) |
11 | tskss 9580 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇 ∧ suc 𝐴 ⊆ 𝒫 𝐴) → suc 𝐴 ∈ 𝑇) | |
12 | 1, 3, 10, 11 | syl3anc 1326 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → suc 𝐴 ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 Ord word 5722 Oncon0 5723 suc csuc 5725 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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-suc 5729 df-tsk 9571 |
This theorem is referenced by: tsk2 9587 |
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