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Theorem ordunisuc 7833
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 7782 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 suceq 6430 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32unieqd 4916 . . . . 5 (𝑥 = 𝐴 suc 𝑥 = suc 𝐴)
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4eqeq12d 2741 . . . 4 (𝑥 = 𝐴 → ( suc 𝑥 = 𝑥 suc 𝐴 = 𝐴))
6 eloni 6374 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
7 ordtr 6378 . . . . . 6 (Ord 𝑥 → Tr 𝑥)
86, 7syl 17 . . . . 5 (𝑥 ∈ On → Tr 𝑥)
9 vex 3467 . . . . . 6 𝑥 ∈ V
109unisuc 6443 . . . . 5 (Tr 𝑥 suc 𝑥 = 𝑥)
118, 10sylib 217 . . . 4 (𝑥 ∈ On → suc 𝑥 = 𝑥)
125, 11vtoclga 3555 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
13 sucon 7804 . . . . . 6 suc On = On
1413unieqi 4915 . . . . 5 suc On = On
15 unon 7832 . . . . 5 On = On
1614, 15eqtri 2753 . . . 4 suc On = On
17 suceq 6430 . . . . 5 (𝐴 = On → suc 𝐴 = suc On)
1817unieqd 4916 . . . 4 (𝐴 = On → suc 𝐴 = suc On)
19 id 22 . . . 4 (𝐴 = On → 𝐴 = On)
2016, 18, 193eqtr4a 2791 . . 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 1533  wcel 2098   cuni 4903  Tr wtr 5260  Ord word 6363  Oncon0 6364  suc csuc 6366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368  df-suc 6370
This theorem is referenced by:  orduniss2  7834  onsucuni2  7835  nlimsucg  7844  tz7.44-2  8426  rnttrcl  9745  ttrclselem2  9749  ttukeylem7  10538  tsksuc  10785  nnuni  35378  dfrdg2  35448  ontgsucval  35973  onsuctopon  35975  limsucncmpi  35986  finxpsuclem  36933
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