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Theorem ordunisuc 7853
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 7803 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 suceq 6449 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32unieqd 4919 . . . . 5 (𝑥 = 𝐴 suc 𝑥 = suc 𝐴)
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4eqeq12d 2752 . . . 4 (𝑥 = 𝐴 → ( suc 𝑥 = 𝑥 suc 𝐴 = 𝐴))
6 eloni 6393 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
7 ordtr 6397 . . . . . 6 (Ord 𝑥 → Tr 𝑥)
86, 7syl 17 . . . . 5 (𝑥 ∈ On → Tr 𝑥)
9 vex 3483 . . . . . 6 𝑥 ∈ V
109unisuc 6462 . . . . 5 (Tr 𝑥 suc 𝑥 = 𝑥)
118, 10sylib 218 . . . 4 (𝑥 ∈ On → suc 𝑥 = 𝑥)
125, 11vtoclga 3576 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
13 sucon 7824 . . . . . 6 suc On = On
1413unieqi 4918 . . . . 5 suc On = On
15 unon 7852 . . . . 5 On = On
1614, 15eqtri 2764 . . . 4 suc On = On
17 suceq 6449 . . . . 5 (𝐴 = On → suc 𝐴 = suc On)
1817unieqd 4919 . . . 4 (𝐴 = On → suc 𝐴 = suc On)
19 id 22 . . . 4 (𝐴 = On → 𝐴 = On)
2016, 18, 193eqtr4a 2802 . . 3 (𝐴 = On → suc 𝐴 = 𝐴)
2112, 20jaoi 857 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → suc 𝐴 = 𝐴)
221, 21sylbi 217 1 (Ord 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 847   = wceq 1539  wcel 2107   cuni 4906  Tr wtr 5258  Ord word 6382  Oncon0 6383  suc csuc 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-suc 6389
This theorem is referenced by:  orduniss2  7854  onsucuni2  7855  nlimsucg  7864  tz7.44-2  8448  rnttrcl  9763  ttrclselem2  9767  ttukeylem7  10556  tsksuc  10803  nnuni  35728  dfrdg2  35797  ontgsucval  36434  onsuctopon  36436  limsucncmpi  36447  finxpsuclem  37399
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