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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 1138 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ Tarski) | |
2 | tskpw 10332 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇) | |
3 | 2 | 3adant2 1133 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇) |
4 | eloni 6201 | . . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
5 | 4 | 3ad2ant2 1136 | . . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → Ord 𝐴) |
6 | ordunisuc 7589 | . . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
7 | eqimss 3943 | . . . 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 4994 | . . 3 ⊢ (suc 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ suc 𝐴 ⊆ 𝐴) | |
10 | 8, 9 | sylibr 237 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → suc 𝐴 ⊆ 𝒫 𝐴) |
11 | tskss 10337 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 𝒫 cpw 4499 ∪ cuni 4805 Ord word 6190 Oncon0 6191 suc csuc 6193 Tarskictsk 10327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-on 6195 df-suc 6197 df-tsk 10328 |
This theorem is referenced by: tsk2 10344 |
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