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Theorem ordunisuc 7852
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 7801 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 suceq 6452 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32unieqd 4925 . . . . 5 (𝑥 = 𝐴 suc 𝑥 = suc 𝐴)
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4eqeq12d 2751 . . . 4 (𝑥 = 𝐴 → ( suc 𝑥 = 𝑥 suc 𝐴 = 𝐴))
6 eloni 6396 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
7 ordtr 6400 . . . . . 6 (Ord 𝑥 → Tr 𝑥)
86, 7syl 17 . . . . 5 (𝑥 ∈ On → Tr 𝑥)
9 vex 3482 . . . . . 6 𝑥 ∈ V
109unisuc 6465 . . . . 5 (Tr 𝑥 suc 𝑥 = 𝑥)
118, 10sylib 218 . . . 4 (𝑥 ∈ On → suc 𝑥 = 𝑥)
125, 11vtoclga 3577 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
13 sucon 7823 . . . . . 6 suc On = On
1413unieqi 4924 . . . . 5 suc On = On
15 unon 7851 . . . . 5 On = On
1614, 15eqtri 2763 . . . 4 suc On = On
17 suceq 6452 . . . . 5 (𝐴 = On → suc 𝐴 = suc On)
1817unieqd 4925 . . . 4 (𝐴 = On → suc 𝐴 = suc On)
19 id 22 . . . 4 (𝐴 = On → 𝐴 = On)
2016, 18, 193eqtr4a 2801 . . 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 1537  wcel 2106   cuni 4912  Tr wtr 5265  Ord word 6385  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  orduniss2  7853  onsucuni2  7854  nlimsucg  7863  tz7.44-2  8446  rnttrcl  9760  ttrclselem2  9764  ttukeylem7  10553  tsksuc  10800  nnuni  35707  dfrdg2  35777  ontgsucval  36415  onsuctopon  36417  limsucncmpi  36428  finxpsuclem  37380
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