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Theorem tsksuc 10757
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 1137 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝑇 ∈ Tarski)
2 tskpw 10748 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
323adant2 1132 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
4 eloni 6375 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
543ad2ant2 1135 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → Ord 𝐴)
6 ordunisuc 7820 . . . 4 (Ord 𝐴 suc 𝐴 = 𝐴)
7 eqimss 4041 . . . 4 ( suc 𝐴 = 𝐴 suc 𝐴𝐴)
85, 6, 73syl 18 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝐴)
9 sspwuni 5104 . . 3 (suc 𝐴 ⊆ 𝒫 𝐴 suc 𝐴𝐴)
108, 9sylibr 233 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴 ⊆ 𝒫 𝐴)
11 tskss 10753 . 2 ((𝑇 ∈ Tarski ∧ 𝒫 𝐴𝑇 ∧ suc 𝐴 ⊆ 𝒫 𝐴) → suc 𝐴𝑇)
121, 3, 10, 11syl3anc 1372 1 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wss 3949  𝒫 cpw 4603   cuni 4909  Ord word 6364  Oncon0 6365  suc csuc 6367  Tarskictsk 10743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-suc 6371  df-tsk 10744
This theorem is referenced by:  tsk2  10760
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