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Theorem ordsucim 4176
Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
Assertion
Ref Expression
ordsucim (Ord A → Ord suc A)

Proof of Theorem ordsucim
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ordtr 4064 . . 3 (Ord A → Tr A)
2 suctr 4108 . . 3 (Tr A → Tr suc A)
31, 2syl 14 . 2 (Ord A → Tr suc A)
4 df-suc 4057 . . . . . 6 suc A = (A ∪ {A})
54eleq2i 2086 . . . . 5 (x suc Ax (A ∪ {A}))
6 elun 3061 . . . . 5 (x (A ∪ {A}) ↔ (x A x {A}))
7 elsn 3365 . . . . . 6 (x {A} ↔ x = A)
87orbi2i 666 . . . . 5 ((x A x {A}) ↔ (x A x = A))
95, 6, 83bitri 195 . . . 4 (x suc A ↔ (x A x = A))
10 dford3 4053 . . . . . . . 8 (Ord A ↔ (Tr A x A Tr x))
1110simprbi 260 . . . . . . 7 (Ord Ax A Tr x)
12 df-ral 2289 . . . . . . 7 (x A Tr xx(x A → Tr x))
1311, 12sylib 127 . . . . . 6 (Ord Ax(x A → Tr x))
141319.21bi 1432 . . . . 5 (Ord A → (x A → Tr x))
15 treq 3834 . . . . . 6 (x = A → (Tr x ↔ Tr A))
161, 15syl5ibrcom 146 . . . . 5 (Ord A → (x = A → Tr x))
1714, 16jaod 624 . . . 4 (Ord A → ((x A x = A) → Tr x))
189, 17syl5bi 141 . . 3 (Ord A → (x suc A → Tr x))
1918ralrimiv 2369 . 2 (Ord Ax suc ATr x)
20 dford3 4053 . 2 (Ord suc A ↔ (Tr suc A x suc ATr x))
213, 19, 20sylanbrc 396 1 (Ord A → Ord suc A)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616  wal 1226   = wceq 1228   wcel 1374  wral 2284  cun 2892  {csn 3350  Tr wtr 3828  Ord word 4048  suc csuc 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-uni 3555  df-tr 3829  df-iord 4052  df-suc 4057
This theorem is referenced by:  suceloni  4177  ordsucg  4178  onsucsssucr  4184  ordtriexmidlem  4192  onsucelsucexmidlem  4198  ordsuc  4225  nnsucsssuc  5986
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