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Theorem ordunisuc 7527
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 7483 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 suceq 6224 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32unieqd 4814 . . . . 5 (𝑥 = 𝐴 suc 𝑥 = suc 𝐴)
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4eqeq12d 2814 . . . 4 (𝑥 = 𝐴 → ( suc 𝑥 = 𝑥 suc 𝐴 = 𝐴))
6 eloni 6169 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
7 ordtr 6173 . . . . . 6 (Ord 𝑥 → Tr 𝑥)
86, 7syl 17 . . . . 5 (𝑥 ∈ On → Tr 𝑥)
9 vex 3444 . . . . . 6 𝑥 ∈ V
109unisuc 6235 . . . . 5 (Tr 𝑥 suc 𝑥 = 𝑥)
118, 10sylib 221 . . . 4 (𝑥 ∈ On → suc 𝑥 = 𝑥)
125, 11vtoclga 3522 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
13 sucon 7503 . . . . . 6 suc On = On
1413unieqi 4813 . . . . 5 suc On = On
15 unon 7526 . . . . 5 On = On
1614, 15eqtri 2821 . . . 4 suc On = On
17 suceq 6224 . . . . 5 (𝐴 = On → suc 𝐴 = suc On)
1817unieqd 4814 . . . 4 (𝐴 = On → suc 𝐴 = suc On)
19 id 22 . . . 4 (𝐴 = On → 𝐴 = On)
2016, 18, 193eqtr4a 2859 . . 3 (𝐴 = On → suc 𝐴 = 𝐴)
2112, 20jaoi 854 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → suc 𝐴 = 𝐴)
221, 21sylbi 220 1 (Ord 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1538  wcel 2111   cuni 4800  Tr wtr 5136  Ord word 6158  Oncon0 6159  suc csuc 6161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-suc 6165
This theorem is referenced by:  orduniss2  7528  onsucuni2  7529  nlimsucg  7537  tz7.44-2  8026  ttukeylem7  9926  tsksuc  10173  dfrdg2  33153  ontgsucval  33893  onsuctopon  33895  limsucncmpi  33906  finxpsuclem  34814
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