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Theorem ordsucim 4192
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 4081 . . 3 (Ord A → Tr A)
2 suctr 4124 . . 3 (Tr A → Tr suc A)
31, 2syl 14 . 2 (Ord A → Tr suc A)
4 df-suc 4074 . . . . . 6 suc A = (A ∪ {A})
54eleq2i 2101 . . . . 5 (x suc Ax (A ∪ {A}))
6 elun 3078 . . . . 5 (x (A ∪ {A}) ↔ (x A x {A}))
7 elsn 3382 . . . . . 6 (x {A} ↔ x = A)
87orbi2i 678 . . . . 5 ((x A x {A}) ↔ (x A x = A))
95, 6, 83bitri 195 . . . 4 (x suc A ↔ (x A x = A))
10 dford3 4070 . . . . . . . 8 (Ord A ↔ (Tr A x A Tr x))
1110simprbi 260 . . . . . . 7 (Ord Ax A Tr x)
12 df-ral 2305 . . . . . . 7 (x A Tr xx(x A → Tr x))
1311, 12sylib 127 . . . . . 6 (Ord Ax(x A → Tr x))
141319.21bi 1447 . . . . 5 (Ord A → (x A → Tr x))
15 treq 3851 . . . . . 6 (x = A → (Tr x ↔ Tr A))
161, 15syl5ibrcom 146 . . . . 5 (Ord A → (x = A → Tr x))
1714, 16jaod 636 . . . 4 (Ord A → ((x A x = A) → Tr x))
189, 17syl5bi 141 . . 3 (Ord A → (x suc A → Tr x))
1918ralrimiv 2385 . 2 (Ord Ax suc ATr x)
20 dford3 4070 . 2 (Ord suc A ↔ (Tr suc A x suc ATr x))
213, 19, 20sylanbrc 394 1 (Ord A → Ord suc A)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628  wal 1240   = wceq 1242   wcel 1390  wral 2300  cun 2909  {csn 3367  Tr wtr 3845  Ord word 4065  suc csuc 4068
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-suc 4074
This theorem is referenced by:  suceloni  4193  ordsucg  4194  onsucsssucr  4200  ordtriexmidlem  4208  onsucelsucexmidlem  4214  ordsuc  4241  nnsucsssuc  6010
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