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Theorem tsksuc 10800
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 ∧ 𝐴𝑇) → suc 𝐴𝑇)

Proof of Theorem tsksuc
StepHypRef Expression
1 simp1 1135 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝑇 ∈ Tarski)
2 tskpw 10791 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
323adant2 1130 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
4 eloni 6396 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
543ad2ant2 1133 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → Ord 𝐴)
6 ordunisuc 7852 . . . 4 (Ord 𝐴 suc 𝐴 = 𝐴)
7 eqimss 4054 . . . 4 ( suc 𝐴 = 𝐴 suc 𝐴𝐴)
85, 6, 73syl 18 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝐴)
9 sspwuni 5105 . . 3 (suc 𝐴 ⊆ 𝒫 𝐴 suc 𝐴𝐴)
108, 9sylibr 234 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴 ⊆ 𝒫 𝐴)
11 tskss 10796 . 2 ((𝑇 ∈ Tarski ∧ 𝒫 𝐴𝑇 ∧ suc 𝐴 ⊆ 𝒫 𝐴) → suc 𝐴𝑇)
121, 3, 10, 11syl3anc 1370 1 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  wss 3963  𝒫 cpw 4605   cuni 4912  Ord word 6385  Oncon0 6386  suc csuc 6388  Tarskictsk 10786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392  df-tsk 10787
This theorem is referenced by:  tsk2  10803
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