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Theorem ordunisuc 7357
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 7313 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 suceq 6088 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32unieqd 4716 . . . . 5 (𝑥 = 𝐴 suc 𝑥 = suc 𝐴)
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4eqeq12d 2787 . . . 4 (𝑥 = 𝐴 → ( suc 𝑥 = 𝑥 suc 𝐴 = 𝐴))
6 eloni 6033 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
7 ordtr 6037 . . . . . 6 (Ord 𝑥 → Tr 𝑥)
86, 7syl 17 . . . . 5 (𝑥 ∈ On → Tr 𝑥)
9 vex 3412 . . . . . 6 𝑥 ∈ V
109unisuc 6099 . . . . 5 (Tr 𝑥 suc 𝑥 = 𝑥)
118, 10sylib 210 . . . 4 (𝑥 ∈ On → suc 𝑥 = 𝑥)
125, 11vtoclga 3487 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
13 sucon 7333 . . . . . 6 suc On = On
1413unieqi 4715 . . . . 5 suc On = On
15 unon 7356 . . . . 5 On = On
1614, 15eqtri 2796 . . . 4 suc On = On
17 suceq 6088 . . . . 5 (𝐴 = On → suc 𝐴 = suc On)
1817unieqd 4716 . . . 4 (𝐴 = On → suc 𝐴 = suc On)
19 id 22 . . . 4 (𝐴 = On → 𝐴 = On)
2016, 18, 193eqtr4a 2834 . . 3 (𝐴 = On → suc 𝐴 = 𝐴)
2112, 20jaoi 843 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → suc 𝐴 = 𝐴)
221, 21sylbi 209 1 (Ord 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 833   = wceq 1507  wcel 2048   cuni 4706  Tr wtr 5024  Ord word 6022  Oncon0 6023  suc csuc 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-tr 5025  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-ord 6026  df-on 6027  df-suc 6029
This theorem is referenced by:  orduniss2  7358  onsucuni2  7359  nlimsucg  7367  tz7.44-2  7840  ttukeylem7  9727  tsksuc  9974  dfrdg2  32501  ontgsucval  33240  onsuctopon  33242  limsucncmpi  33253  finxpsuclem  34054
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