MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tsksuc Structured version   Visualization version   GIF version

Theorem tsksuc 10177
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 1133 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝑇 ∈ Tarski)
2 tskpw 10168 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
323adant2 1128 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
4 eloni 6173 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
543ad2ant2 1131 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → Ord 𝐴)
6 ordunisuc 7531 . . . 4 (Ord 𝐴 suc 𝐴 = 𝐴)
7 eqimss 3974 . . . 4 ( suc 𝐴 = 𝐴 suc 𝐴𝐴)
85, 6, 73syl 18 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝐴)
9 sspwuni 4988 . . 3 (suc 𝐴 ⊆ 𝒫 𝐴 suc 𝐴𝐴)
108, 9sylibr 237 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴 ⊆ 𝒫 𝐴)
11 tskss 10173 . 2 ((𝑇 ∈ Tarski ∧ 𝒫 𝐴𝑇 ∧ suc 𝐴 ⊆ 𝒫 𝐴) → suc 𝐴𝑇)
121, 3, 10, 11syl3anc 1368 1 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2112  wss 3884  𝒫 cpw 4500   cuni 4803  Ord word 6162  Oncon0 6163  suc csuc 6165  Tarskictsk 10163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167  df-suc 6169  df-tsk 10164
This theorem is referenced by:  tsk2  10180
  Copyright terms: Public domain W3C validator