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Theorem ordsucim 4345
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 4229 . . 3 (Ord 𝐴 → Tr 𝐴)
2 suctr 4272 . . 3 (Tr 𝐴 → Tr suc 𝐴)
31, 2syl 14 . 2 (Ord 𝐴 → Tr suc 𝐴)
4 df-suc 4222 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
54eleq2i 2161 . . . . 5 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
6 elun 3156 . . . . 5 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
7 velsn 3483 . . . . . 6 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
87orbi2i 717 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ (𝑥𝐴𝑥 = 𝐴))
95, 6, 83bitri 205 . . . 4 (𝑥 ∈ suc 𝐴 ↔ (𝑥𝐴𝑥 = 𝐴))
10 dford3 4218 . . . . . . . 8 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
1110simprbi 270 . . . . . . 7 (Ord 𝐴 → ∀𝑥𝐴 Tr 𝑥)
12 df-ral 2375 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥(𝑥𝐴 → Tr 𝑥))
1311, 12sylib 121 . . . . . 6 (Ord 𝐴 → ∀𝑥(𝑥𝐴 → Tr 𝑥))
141319.21bi 1502 . . . . 5 (Ord 𝐴 → (𝑥𝐴 → Tr 𝑥))
15 treq 3964 . . . . . 6 (𝑥 = 𝐴 → (Tr 𝑥 ↔ Tr 𝐴))
161, 15syl5ibrcom 156 . . . . 5 (Ord 𝐴 → (𝑥 = 𝐴 → Tr 𝑥))
1714, 16jaod 675 . . . 4 (Ord 𝐴 → ((𝑥𝐴𝑥 = 𝐴) → Tr 𝑥))
189, 17syl5bi 151 . . 3 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → Tr 𝑥))
1918ralrimiv 2457 . 2 (Ord 𝐴 → ∀𝑥 ∈ suc 𝐴Tr 𝑥)
20 dford3 4218 . 2 (Ord suc 𝐴 ↔ (Tr suc 𝐴 ∧ ∀𝑥 ∈ suc 𝐴Tr 𝑥))
213, 19, 20sylanbrc 409 1 (Ord 𝐴 → Ord suc 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 667  wal 1294   = wceq 1296  wcel 1445  wral 2370  cun 3011  {csn 3466  Tr wtr 3958  Ord word 4213  suc csuc 4216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-uni 3676  df-tr 3959  df-iord 4217  df-suc 4222
This theorem is referenced by:  suceloni  4346  ordsucg  4347  onsucsssucr  4354  ordtriexmidlem  4364  2ordpr  4368  ordsuc  4407  nnsucsssuc  6293
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