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Theorem ordsuc 4369
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 4307 . 2 (Ord 𝐴 → Ord suc 𝐴)
2 en2lp 4360 . . . . . . . . . 10 ¬ (𝑥𝐴𝐴𝑥)
3 eleq1 2150 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
43biimpac 292 . . . . . . . . . . . 12 ((𝑦𝑥𝑦 = 𝐴) → 𝐴𝑥)
54anim2i 334 . . . . . . . . . . 11 ((𝑥𝐴 ∧ (𝑦𝑥𝑦 = 𝐴)) → (𝑥𝐴𝐴𝑥))
65expr 367 . . . . . . . . . 10 ((𝑥𝐴𝑦𝑥) → (𝑦 = 𝐴 → (𝑥𝐴𝐴𝑥)))
72, 6mtoi 625 . . . . . . . . 9 ((𝑥𝐴𝑦𝑥) → ¬ 𝑦 = 𝐴)
87adantl 271 . . . . . . . 8 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → ¬ 𝑦 = 𝐴)
9 elelsuc 4227 . . . . . . . . . . . . . . 15 (𝑥𝐴𝑥 ∈ suc 𝐴)
109adantr 270 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑦𝑥) → 𝑥 ∈ suc 𝐴)
11 ordelss 4197 . . . . . . . . . . . . . 14 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ⊆ suc 𝐴)
1210, 11sylan2 280 . . . . . . . . . . . . 13 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑥 ⊆ suc 𝐴)
1312sseld 3022 . . . . . . . . . . . 12 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → (𝑦𝑥𝑦 ∈ suc 𝐴))
1413expr 367 . . . . . . . . . . 11 ((Ord suc 𝐴𝑥𝐴) → (𝑦𝑥 → (𝑦𝑥𝑦 ∈ suc 𝐴)))
1514pm2.43d 49 . . . . . . . . . 10 ((Ord suc 𝐴𝑥𝐴) → (𝑦𝑥𝑦 ∈ suc 𝐴))
1615impr 371 . . . . . . . . 9 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑦 ∈ suc 𝐴)
17 elsuci 4221 . . . . . . . . 9 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
1816, 17syl 14 . . . . . . . 8 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → (𝑦𝐴𝑦 = 𝐴))
198, 18ecased 1285 . . . . . . 7 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑦𝐴)
2019ancom2s 533 . . . . . 6 ((Ord suc 𝐴 ∧ (𝑦𝑥𝑥𝐴)) → 𝑦𝐴)
2120ex 113 . . . . 5 (Ord suc 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2221alrimivv 1803 . . . 4 (Ord suc 𝐴 → ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
23 dftr2 3930 . . . 4 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2422, 23sylibr 132 . . 3 (Ord suc 𝐴 → Tr 𝐴)
25 sssucid 4233 . . . 4 𝐴 ⊆ suc 𝐴
26 trssord 4198 . . . 4 ((Tr 𝐴𝐴 ⊆ suc 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
2725, 26mp3an2 1261 . . 3 ((Tr 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
2824, 27mpancom 413 . 2 (Ord suc 𝐴 → Ord 𝐴)
291, 28impbii 124 1 (Ord 𝐴 ↔ Ord suc 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  wal 1287   = wceq 1289  wcel 1438  wss 2997  Tr wtr 3928  Ord word 4180  suc csuc 4183
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-uni 3649  df-tr 3929  df-iord 4184  df-suc 4189
This theorem is referenced by:  nlimsucg  4372  ordpwsucss  4373
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