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Theorem ordsucim 4514
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 4393 . . 3 (Ord 𝐴 → Tr 𝐴)
2 suctr 4436 . . 3 (Tr 𝐴 → Tr suc 𝐴)
31, 2syl 14 . 2 (Ord 𝐴 → Tr suc 𝐴)
4 df-suc 4386 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
54eleq2i 2256 . . . . 5 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
6 elun 3291 . . . . 5 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
7 velsn 3624 . . . . . 6 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
87orbi2i 763 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ (𝑥𝐴𝑥 = 𝐴))
95, 6, 83bitri 206 . . . 4 (𝑥 ∈ suc 𝐴 ↔ (𝑥𝐴𝑥 = 𝐴))
10 dford3 4382 . . . . . . . 8 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
1110simprbi 275 . . . . . . 7 (Ord 𝐴 → ∀𝑥𝐴 Tr 𝑥)
12 df-ral 2473 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥(𝑥𝐴 → Tr 𝑥))
1311, 12sylib 122 . . . . . 6 (Ord 𝐴 → ∀𝑥(𝑥𝐴 → Tr 𝑥))
141319.21bi 1569 . . . . 5 (Ord 𝐴 → (𝑥𝐴 → Tr 𝑥))
15 treq 4122 . . . . . 6 (𝑥 = 𝐴 → (Tr 𝑥 ↔ Tr 𝐴))
161, 15syl5ibrcom 157 . . . . 5 (Ord 𝐴 → (𝑥 = 𝐴 → Tr 𝑥))
1714, 16jaod 718 . . . 4 (Ord 𝐴 → ((𝑥𝐴𝑥 = 𝐴) → Tr 𝑥))
189, 17biimtrid 152 . . 3 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → Tr 𝑥))
1918ralrimiv 2562 . 2 (Ord 𝐴 → ∀𝑥 ∈ suc 𝐴Tr 𝑥)
20 dford3 4382 . 2 (Ord suc 𝐴 ↔ (Tr suc 𝐴 ∧ ∀𝑥 ∈ suc 𝐴Tr 𝑥))
213, 19, 20sylanbrc 417 1 (Ord 𝐴 → Ord suc 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 709  wal 1362   = wceq 1364  wcel 2160  wral 2468  cun 3142  {csn 3607  Tr wtr 4116  Ord word 4377  suc csuc 4380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-uni 3825  df-tr 4117  df-iord 4381  df-suc 4386
This theorem is referenced by:  onsuc  4515  ordsucg  4516  onsucsssucr  4523  ordtriexmidlem  4533  2ordpr  4538  ordsuc  4577  nnsucsssuc  6511
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