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

Theorem ordunisuc 7816
Description: An ordinal class is equal to the union of its successor. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ordunisuc (Ord 𝐴 suc 𝐴 = 𝐴)

Proof of Theorem ordunisuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ordeleqon 7765 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 suceq 6427 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32unieqd 4921 . . . . 5 (𝑥 = 𝐴 suc 𝑥 = suc 𝐴)
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4eqeq12d 2748 . . . 4 (𝑥 = 𝐴 → ( suc 𝑥 = 𝑥 suc 𝐴 = 𝐴))
6 eloni 6371 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
7 ordtr 6375 . . . . . 6 (Ord 𝑥 → Tr 𝑥)
86, 7syl 17 . . . . 5 (𝑥 ∈ On → Tr 𝑥)
9 vex 3478 . . . . . 6 𝑥 ∈ V
109unisuc 6440 . . . . 5 (Tr 𝑥 suc 𝑥 = 𝑥)
118, 10sylib 217 . . . 4 (𝑥 ∈ On → suc 𝑥 = 𝑥)
125, 11vtoclga 3565 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
13 sucon 7787 . . . . . 6 suc On = On
1413unieqi 4920 . . . . 5 suc On = On
15 unon 7815 . . . . 5 On = On
1614, 15eqtri 2760 . . . 4 suc On = On
17 suceq 6427 . . . . 5 (𝐴 = On → suc 𝐴 = suc On)
1817unieqd 4921 . . . 4 (𝐴 = On → suc 𝐴 = suc On)
19 id 22 . . . 4 (𝐴 = On → 𝐴 = On)
2016, 18, 193eqtr4a 2798 . . 3 (𝐴 = On → suc 𝐴 = 𝐴)
2112, 20jaoi 855 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → suc 𝐴 = 𝐴)
221, 21sylbi 216 1 (Ord 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 845   = wceq 1541  wcel 2106   cuni 4907  Tr wtr 5264  Ord word 6360  Oncon0 6361  suc csuc 6363
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-suc 6367
This theorem is referenced by:  orduniss2  7817  onsucuni2  7818  nlimsucg  7827  tz7.44-2  8403  rnttrcl  9713  ttrclselem2  9717  ttukeylem7  10506  tsksuc  10753  nnuni  34684  dfrdg2  34755  ontgsucval  35305  onsuctopon  35307  limsucncmpi  35318  finxpsuclem  36266
  Copyright terms: Public domain W3C validator