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Theorem ordunisuc 7180
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 7136 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 suceq 5934 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32unieqd 4585 . . . . 5 (𝑥 = 𝐴 suc 𝑥 = suc 𝐴)
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4eqeq12d 2786 . . . 4 (𝑥 = 𝐴 → ( suc 𝑥 = 𝑥 suc 𝐴 = 𝐴))
6 eloni 5877 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
7 ordtr 5881 . . . . . 6 (Ord 𝑥 → Tr 𝑥)
86, 7syl 17 . . . . 5 (𝑥 ∈ On → Tr 𝑥)
9 vex 3354 . . . . . 6 𝑥 ∈ V
109unisuc 5945 . . . . 5 (Tr 𝑥 suc 𝑥 = 𝑥)
118, 10sylib 208 . . . 4 (𝑥 ∈ On → suc 𝑥 = 𝑥)
125, 11vtoclga 3424 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
13 sucon 7156 . . . . . 6 suc On = On
1413unieqi 4584 . . . . 5 suc On = On
15 unon 7179 . . . . 5 On = On
1614, 15eqtri 2793 . . . 4 suc On = On
17 suceq 5934 . . . . 5 (𝐴 = On → suc 𝐴 = suc On)
1817unieqd 4585 . . . 4 (𝐴 = On → suc 𝐴 = suc On)
19 id 22 . . . 4 (𝐴 = On → 𝐴 = On)
2016, 18, 193eqtr4a 2831 . . 3 (𝐴 = On → suc 𝐴 = 𝐴)
2112, 20jaoi 838 . 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 828   = wceq 1631  wcel 2145   cuni 4575  Tr wtr 4887  Ord word 5866  Oncon0 5867  suc csuc 5869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5870  df-on 5871  df-suc 5873
This theorem is referenced by:  orduniss2  7181  onsucuni2  7182  nlimsucg  7190  tz7.44-2  7657  ttukeylem7  9540  tsksuc  9787  dfrdg2  32038  ontgsucval  32769  onsuctopon  32771  limsucncmpi  32782  finxpsuclem  33572
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