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Theorem ordsucim 4416
 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 4300 . . 3 (Ord 𝐴 → Tr 𝐴)
2 suctr 4343 . . 3 (Tr 𝐴 → Tr suc 𝐴)
31, 2syl 14 . 2 (Ord 𝐴 → Tr suc 𝐴)
4 df-suc 4293 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
54eleq2i 2206 . . . . 5 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
6 elun 3217 . . . . 5 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
7 velsn 3544 . . . . . 6 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
87orbi2i 751 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ (𝑥𝐴𝑥 = 𝐴))
95, 6, 83bitri 205 . . . 4 (𝑥 ∈ suc 𝐴 ↔ (𝑥𝐴𝑥 = 𝐴))
10 dford3 4289 . . . . . . . 8 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
1110simprbi 273 . . . . . . 7 (Ord 𝐴 → ∀𝑥𝐴 Tr 𝑥)
12 df-ral 2421 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥(𝑥𝐴 → Tr 𝑥))
1311, 12sylib 121 . . . . . 6 (Ord 𝐴 → ∀𝑥(𝑥𝐴 → Tr 𝑥))
141319.21bi 1537 . . . . 5 (Ord 𝐴 → (𝑥𝐴 → Tr 𝑥))
15 treq 4032 . . . . . 6 (𝑥 = 𝐴 → (Tr 𝑥 ↔ Tr 𝐴))
161, 15syl5ibrcom 156 . . . . 5 (Ord 𝐴 → (𝑥 = 𝐴 → Tr 𝑥))
1714, 16jaod 706 . . . 4 (Ord 𝐴 → ((𝑥𝐴𝑥 = 𝐴) → Tr 𝑥))
189, 17syl5bi 151 . . 3 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → Tr 𝑥))
1918ralrimiv 2504 . 2 (Ord 𝐴 → ∀𝑥 ∈ suc 𝐴Tr 𝑥)
20 dford3 4289 . 2 (Ord suc 𝐴 ↔ (Tr suc 𝐴 ∧ ∀𝑥 ∈ suc 𝐴Tr 𝑥))
213, 19, 20sylanbrc 413 1 (Ord 𝐴 → Ord suc 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 697  ∀wal 1329   = wceq 1331   ∈ wcel 1480  ∀wral 2416   ∪ cun 3069  {csn 3527  Tr wtr 4026  Ord word 4284  suc csuc 4287 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-uni 3737  df-tr 4027  df-iord 4288  df-suc 4293 This theorem is referenced by:  suceloni  4417  ordsucg  4418  onsucsssucr  4425  ordtriexmidlem  4435  2ordpr  4439  ordsuc  4478  nnsucsssuc  6388
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