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

Theorem tsksuc 9776
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 1131 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝑇 ∈ Tarski)
2 tskpw 9767 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
323adant2 1126 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
4 eloni 5894 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
543ad2ant2 1129 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → Ord 𝐴)
6 ordunisuc 7197 . . . 4 (Ord 𝐴 suc 𝐴 = 𝐴)
7 eqimss 3798 . . . 4 ( suc 𝐴 = 𝐴 suc 𝐴𝐴)
85, 6, 73syl 18 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝐴)
9 sspwuni 4763 . . 3 (suc 𝐴 ⊆ 𝒫 𝐴 suc 𝐴𝐴)
108, 9sylibr 224 . 2 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴 ⊆ 𝒫 𝐴)
11 tskss 9772 . 2 ((𝑇 ∈ Tarski ∧ 𝒫 𝐴𝑇 ∧ suc 𝐴 ⊆ 𝒫 𝐴) → suc 𝐴𝑇)
121, 3, 10, 11syl3anc 1477 1 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139  wss 3715  𝒫 cpw 4302   cuni 4588  Ord word 5883  Oncon0 5884  suc csuc 5886  Tarskictsk 9762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888  df-suc 5890  df-tsk 9763
This theorem is referenced by:  tsk2  9779
  Copyright terms: Public domain W3C validator