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Theorem ordunisuc 7817
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 7766 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 suceq 6424 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32unieqd 4915 . . . . 5 (𝑥 = 𝐴 suc 𝑥 = suc 𝐴)
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4eqeq12d 2742 . . . 4 (𝑥 = 𝐴 → ( suc 𝑥 = 𝑥 suc 𝐴 = 𝐴))
6 eloni 6368 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
7 ordtr 6372 . . . . . 6 (Ord 𝑥 → Tr 𝑥)
86, 7syl 17 . . . . 5 (𝑥 ∈ On → Tr 𝑥)
9 vex 3472 . . . . . 6 𝑥 ∈ V
109unisuc 6437 . . . . 5 (Tr 𝑥 suc 𝑥 = 𝑥)
118, 10sylib 217 . . . 4 (𝑥 ∈ On → suc 𝑥 = 𝑥)
125, 11vtoclga 3560 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
13 sucon 7788 . . . . . 6 suc On = On
1413unieqi 4914 . . . . 5 suc On = On
15 unon 7816 . . . . 5 On = On
1614, 15eqtri 2754 . . . 4 suc On = On
17 suceq 6424 . . . . 5 (𝐴 = On → suc 𝐴 = suc On)
1817unieqd 4915 . . . 4 (𝐴 = On → suc 𝐴 = suc On)
19 id 22 . . . 4 (𝐴 = On → 𝐴 = On)
2016, 18, 193eqtr4a 2792 . . 3 (𝐴 = On → suc 𝐴 = 𝐴)
2112, 20jaoi 854 . 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 844   = wceq 1533  wcel 2098   cuni 4902  Tr wtr 5258  Ord word 6357  Oncon0 6358  suc csuc 6360
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6361  df-on 6362  df-suc 6364
This theorem is referenced by:  orduniss2  7818  onsucuni2  7819  nlimsucg  7828  tz7.44-2  8408  rnttrcl  9719  ttrclselem2  9723  ttukeylem7  10512  tsksuc  10759  nnuni  35230  dfrdg2  35300  ontgsucval  35825  onsuctopon  35827  limsucncmpi  35838  finxpsuclem  36785
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