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Theorem ordsucim 4253
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 4142 . . 3 (Ord 𝐴 → Tr 𝐴)
2 suctr 4185 . . 3 (Tr 𝐴 → Tr suc 𝐴)
31, 2syl 14 . 2 (Ord 𝐴 → Tr suc 𝐴)
4 df-suc 4135 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
54eleq2i 2120 . . . . 5 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
6 elun 3111 . . . . 5 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
7 velsn 3419 . . . . . 6 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
87orbi2i 689 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ (𝑥𝐴𝑥 = 𝐴))
95, 6, 83bitri 199 . . . 4 (𝑥 ∈ suc 𝐴 ↔ (𝑥𝐴𝑥 = 𝐴))
10 dford3 4131 . . . . . . . 8 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
1110simprbi 264 . . . . . . 7 (Ord 𝐴 → ∀𝑥𝐴 Tr 𝑥)
12 df-ral 2328 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥(𝑥𝐴 → Tr 𝑥))
1311, 12sylib 131 . . . . . 6 (Ord 𝐴 → ∀𝑥(𝑥𝐴 → Tr 𝑥))
141319.21bi 1466 . . . . 5 (Ord 𝐴 → (𝑥𝐴 → Tr 𝑥))
15 treq 3887 . . . . . 6 (𝑥 = 𝐴 → (Tr 𝑥 ↔ Tr 𝐴))
161, 15syl5ibrcom 150 . . . . 5 (Ord 𝐴 → (𝑥 = 𝐴 → Tr 𝑥))
1714, 16jaod 647 . . . 4 (Ord 𝐴 → ((𝑥𝐴𝑥 = 𝐴) → Tr 𝑥))
189, 17syl5bi 145 . . 3 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → Tr 𝑥))
1918ralrimiv 2408 . 2 (Ord 𝐴 → ∀𝑥 ∈ suc 𝐴Tr 𝑥)
20 dford3 4131 . 2 (Ord suc 𝐴 ↔ (Tr suc 𝐴 ∧ ∀𝑥 ∈ suc 𝐴Tr 𝑥))
213, 19, 20sylanbrc 402 1 (Ord 𝐴 → Ord suc 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 639  wal 1257   = wceq 1259  wcel 1409  wral 2323  cun 2942  {csn 3402  Tr wtr 3881  Ord word 4126  suc csuc 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-uni 3608  df-tr 3882  df-iord 4130  df-suc 4135
This theorem is referenced by:  suceloni  4254  ordsucg  4255  onsucsssucr  4262  ordtriexmidlem  4272  2ordpr  4276  ordsuc  4314  nnsucsssuc  6101
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