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Theorem ordunisuc 6924
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 6880 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 suceq 5707 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32unieqd 4382 . . . . 5 (𝑥 = 𝐴 suc 𝑥 = suc 𝐴)
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4eqeq12d 2625 . . . 4 (𝑥 = 𝐴 → ( suc 𝑥 = 𝑥 suc 𝐴 = 𝐴))
6 eloni 5650 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
7 ordtr 5654 . . . . . 6 (Ord 𝑥 → Tr 𝑥)
86, 7syl 17 . . . . 5 (𝑥 ∈ On → Tr 𝑥)
9 vex 3176 . . . . . 6 𝑥 ∈ V
109unisuc 5718 . . . . 5 (Tr 𝑥 suc 𝑥 = 𝑥)
118, 10sylib 207 . . . 4 (𝑥 ∈ On → suc 𝑥 = 𝑥)
125, 11vtoclga 3245 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
13 sucon 6900 . . . . . 6 suc On = On
1413unieqi 4381 . . . . 5 suc On = On
15 unon 6923 . . . . 5 On = On
1614, 15eqtri 2632 . . . 4 suc On = On
17 suceq 5707 . . . . 5 (𝐴 = On → suc 𝐴 = suc On)
1817unieqd 4382 . . . 4 (𝐴 = On → suc 𝐴 = suc On)
19 id 22 . . . 4 (𝐴 = On → 𝐴 = On)
2016, 18, 193eqtr4a 2670 . . 3 (𝐴 = On → suc 𝐴 = 𝐴)
2112, 20jaoi 393 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → suc 𝐴 = 𝐴)
221, 21sylbi 206 1 (Ord 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382   = wceq 1475  wcel 1977   cuni 4372  Tr wtr 4680  Ord word 5639  Oncon0 5640  suc csuc 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646
This theorem is referenced by:  orduniss2  6925  onsucuni2  6926  nlimsucg  6934  tz7.44-2  7390  ttukeylem7  9220  tsksuc  9463  dfrdg2  30945  ontgsucval  31601  onsuctopon  31603  limsucncmpi  31614  finxpsuclem  32410
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