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

Theorem ordsuc 4314
Description: The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.)
Assertion
Ref Expression
ordsuc (Ord 𝐴 ↔ Ord suc 𝐴)

Proof of Theorem ordsuc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordsucim 4253 . 2 (Ord 𝐴 → Ord suc 𝐴)
2 en2lp 4305 . . . . . . . . . 10 ¬ (𝑥𝐴𝐴𝑥)
3 eleq1 2116 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
43biimpac 286 . . . . . . . . . . . 12 ((𝑦𝑥𝑦 = 𝐴) → 𝐴𝑥)
54anim2i 328 . . . . . . . . . . 11 ((𝑥𝐴 ∧ (𝑦𝑥𝑦 = 𝐴)) → (𝑥𝐴𝐴𝑥))
65expr 361 . . . . . . . . . 10 ((𝑥𝐴𝑦𝑥) → (𝑦 = 𝐴 → (𝑥𝐴𝐴𝑥)))
72, 6mtoi 600 . . . . . . . . 9 ((𝑥𝐴𝑦𝑥) → ¬ 𝑦 = 𝐴)
87adantl 266 . . . . . . . 8 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → ¬ 𝑦 = 𝐴)
9 elelsuc 4173 . . . . . . . . . . . . . . 15 (𝑥𝐴𝑥 ∈ suc 𝐴)
109adantr 265 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑦𝑥) → 𝑥 ∈ suc 𝐴)
11 ordelss 4143 . . . . . . . . . . . . . 14 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ⊆ suc 𝐴)
1210, 11sylan2 274 . . . . . . . . . . . . 13 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑥 ⊆ suc 𝐴)
1312sseld 2971 . . . . . . . . . . . 12 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → (𝑦𝑥𝑦 ∈ suc 𝐴))
1413expr 361 . . . . . . . . . . 11 ((Ord suc 𝐴𝑥𝐴) → (𝑦𝑥 → (𝑦𝑥𝑦 ∈ suc 𝐴)))
1514pm2.43d 48 . . . . . . . . . 10 ((Ord suc 𝐴𝑥𝐴) → (𝑦𝑥𝑦 ∈ suc 𝐴))
1615impr 365 . . . . . . . . 9 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑦 ∈ suc 𝐴)
17 elsuci 4167 . . . . . . . . 9 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
1816, 17syl 14 . . . . . . . 8 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → (𝑦𝐴𝑦 = 𝐴))
198, 18ecased 1255 . . . . . . 7 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑦𝐴)
2019ancom2s 508 . . . . . 6 ((Ord suc 𝐴 ∧ (𝑦𝑥𝑥𝐴)) → 𝑦𝐴)
2120ex 112 . . . . 5 (Ord suc 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2221alrimivv 1771 . . . 4 (Ord suc 𝐴 → ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
23 dftr2 3883 . . . 4 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2422, 23sylibr 141 . . 3 (Ord suc 𝐴 → Tr 𝐴)
25 sssucid 4179 . . . 4 𝐴 ⊆ suc 𝐴
26 trssord 4144 . . . 4 ((Tr 𝐴𝐴 ⊆ suc 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
2725, 26mp3an2 1231 . . 3 ((Tr 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
2824, 27mpancom 407 . 2 (Ord suc 𝐴 → Ord 𝐴)
291, 28impbii 121 1 (Ord 𝐴 ↔ Ord suc 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  wal 1257   = wceq 1259  wcel 1409  wss 2944  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-in1 554  ax-in2 555  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  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  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-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-uni 3608  df-tr 3882  df-iord 4130  df-suc 4135
This theorem is referenced by:  nlimsucg  4317  ordpwsucss  4318
  Copyright terms: Public domain W3C validator