ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordsucim GIF version

Theorem ordsucim 4290
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 4179 . . 3 (Ord 𝐴 → Tr 𝐴)
2 suctr 4222 . . 3 (Tr 𝐴 → Tr suc 𝐴)
31, 2syl 14 . 2 (Ord 𝐴 → Tr suc 𝐴)
4 df-suc 4172 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
54eleq2i 2151 . . . . 5 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
6 elun 3130 . . . . 5 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
7 velsn 3448 . . . . . 6 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
87orbi2i 712 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ (𝑥𝐴𝑥 = 𝐴))
95, 6, 83bitri 204 . . . 4 (𝑥 ∈ suc 𝐴 ↔ (𝑥𝐴𝑥 = 𝐴))
10 dford3 4168 . . . . . . . 8 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
1110simprbi 269 . . . . . . 7 (Ord 𝐴 → ∀𝑥𝐴 Tr 𝑥)
12 df-ral 2360 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥(𝑥𝐴 → Tr 𝑥))
1311, 12sylib 120 . . . . . 6 (Ord 𝐴 → ∀𝑥(𝑥𝐴 → Tr 𝑥))
141319.21bi 1493 . . . . 5 (Ord 𝐴 → (𝑥𝐴 → Tr 𝑥))
15 treq 3917 . . . . . 6 (𝑥 = 𝐴 → (Tr 𝑥 ↔ Tr 𝐴))
161, 15syl5ibrcom 155 . . . . 5 (Ord 𝐴 → (𝑥 = 𝐴 → Tr 𝑥))
1714, 16jaod 670 . . . 4 (Ord 𝐴 → ((𝑥𝐴𝑥 = 𝐴) → Tr 𝑥))
189, 17syl5bi 150 . . 3 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → Tr 𝑥))
1918ralrimiv 2441 . 2 (Ord 𝐴 → ∀𝑥 ∈ suc 𝐴Tr 𝑥)
20 dford3 4168 . 2 (Ord suc 𝐴 ↔ (Tr suc 𝐴 ∧ ∀𝑥 ∈ suc 𝐴Tr 𝑥))
213, 19, 20sylanbrc 408 1 (Ord 𝐴 → Ord suc 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662  wal 1285   = wceq 1287  wcel 1436  wral 2355  cun 2986  {csn 3431  Tr wtr 3911  Ord word 4163  suc csuc 4166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-sn 3437  df-uni 3637  df-tr 3912  df-iord 4167  df-suc 4172
This theorem is referenced by:  suceloni  4291  ordsucg  4292  onsucsssucr  4299  ordtriexmidlem  4309  2ordpr  4313  ordsuc  4352  nnsucsssuc  6207
  Copyright terms: Public domain W3C validator