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Theorem ordunisuc 7074
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 7030 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 suceq 5828 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32unieqd 4478 . . . . 5 (𝑥 = 𝐴 suc 𝑥 = suc 𝐴)
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4eqeq12d 2666 . . . 4 (𝑥 = 𝐴 → ( suc 𝑥 = 𝑥 suc 𝐴 = 𝐴))
6 eloni 5771 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
7 ordtr 5775 . . . . . 6 (Ord 𝑥 → Tr 𝑥)
86, 7syl 17 . . . . 5 (𝑥 ∈ On → Tr 𝑥)
9 vex 3234 . . . . . 6 𝑥 ∈ V
109unisuc 5839 . . . . 5 (Tr 𝑥 suc 𝑥 = 𝑥)
118, 10sylib 208 . . . 4 (𝑥 ∈ On → suc 𝑥 = 𝑥)
125, 11vtoclga 3303 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
13 sucon 7050 . . . . . 6 suc On = On
1413unieqi 4477 . . . . 5 suc On = On
15 unon 7073 . . . . 5 On = On
1614, 15eqtri 2673 . . . 4 suc On = On
17 suceq 5828 . . . . 5 (𝐴 = On → suc 𝐴 = suc On)
1817unieqd 4478 . . . 4 (𝐴 = On → suc 𝐴 = suc On)
19 id 22 . . . 4 (𝐴 = On → 𝐴 = On)
2016, 18, 193eqtr4a 2711 . . 3 (𝐴 = On → suc 𝐴 = 𝐴)
2112, 20jaoi 393 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → suc 𝐴 = 𝐴)
221, 21sylbi 207 1 (Ord 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382   = wceq 1523  wcel 2030   cuni 4468  Tr wtr 4785  Ord word 5760  Oncon0 5761  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767
This theorem is referenced by:  orduniss2  7075  onsucuni2  7076  nlimsucg  7084  tz7.44-2  7548  ttukeylem7  9375  tsksuc  9622  dfrdg2  31825  ontgsucval  32556  onsuctopon  32558  limsucncmpi  32569  finxpsuclem  33364
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