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Theorem tsksuc 10172
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 1128 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝑇 ∈ Tarski)
2 tskpw 10163 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
323adant2 1123 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
4 eloni 6194 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
543ad2ant2 1126 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → Ord 𝐴)
6 ordunisuc 7536 . . . 4 (Ord 𝐴 suc 𝐴 = 𝐴)
7 eqimss 4020 . . . 4 ( suc 𝐴 = 𝐴 suc 𝐴𝐴)
85, 6, 73syl 18 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝐴)
9 sspwuni 5013 . . 3 (suc 𝐴 ⊆ 𝒫 𝐴 suc 𝐴𝐴)
108, 9sylibr 235 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴 ⊆ 𝒫 𝐴)
11 tskss 10168 . 2 ((𝑇 ∈ Tarski ∧ 𝒫 𝐴𝑇 ∧ suc 𝐴 ⊆ 𝒫 𝐴) → suc 𝐴𝑇)
121, 3, 10, 11syl3anc 1363 1 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  wss 3933  𝒫 cpw 4535   cuni 4830  Ord word 6183  Oncon0 6184  suc csuc 6186  Tarskictsk 10158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-suc 6190  df-tsk 10159
This theorem is referenced by:  tsk2  10175
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