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Theorem ordsuc 4580
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 4517 . 2 (Ord 𝐴 → Ord suc 𝐴)
2 en2lp 4571 . . . . . . . . . 10 ¬ (𝑥𝐴𝐴𝑥)
3 eleq1 2252 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
43biimpac 298 . . . . . . . . . . . 12 ((𝑦𝑥𝑦 = 𝐴) → 𝐴𝑥)
54anim2i 342 . . . . . . . . . . 11 ((𝑥𝐴 ∧ (𝑦𝑥𝑦 = 𝐴)) → (𝑥𝐴𝐴𝑥))
65expr 375 . . . . . . . . . 10 ((𝑥𝐴𝑦𝑥) → (𝑦 = 𝐴 → (𝑥𝐴𝐴𝑥)))
72, 6mtoi 665 . . . . . . . . 9 ((𝑥𝐴𝑦𝑥) → ¬ 𝑦 = 𝐴)
87adantl 277 . . . . . . . 8 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → ¬ 𝑦 = 𝐴)
9 elelsuc 4427 . . . . . . . . . . . . . . 15 (𝑥𝐴𝑥 ∈ suc 𝐴)
109adantr 276 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑦𝑥) → 𝑥 ∈ suc 𝐴)
11 ordelss 4397 . . . . . . . . . . . . . 14 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ⊆ suc 𝐴)
1210, 11sylan2 286 . . . . . . . . . . . . 13 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑥 ⊆ suc 𝐴)
1312sseld 3169 . . . . . . . . . . . 12 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → (𝑦𝑥𝑦 ∈ suc 𝐴))
1413expr 375 . . . . . . . . . . 11 ((Ord suc 𝐴𝑥𝐴) → (𝑦𝑥 → (𝑦𝑥𝑦 ∈ suc 𝐴)))
1514pm2.43d 50 . . . . . . . . . 10 ((Ord suc 𝐴𝑥𝐴) → (𝑦𝑥𝑦 ∈ suc 𝐴))
1615impr 379 . . . . . . . . 9 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑦 ∈ suc 𝐴)
17 elsuci 4421 . . . . . . . . 9 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
1816, 17syl 14 . . . . . . . 8 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → (𝑦𝐴𝑦 = 𝐴))
198, 18ecased 1360 . . . . . . 7 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑦𝐴)
2019ancom2s 566 . . . . . 6 ((Ord suc 𝐴 ∧ (𝑦𝑥𝑥𝐴)) → 𝑦𝐴)
2120ex 115 . . . . 5 (Ord suc 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2221alrimivv 1886 . . . 4 (Ord suc 𝐴 → ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
23 dftr2 4118 . . . 4 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2422, 23sylibr 134 . . 3 (Ord suc 𝐴 → Tr 𝐴)
25 sssucid 4433 . . . 4 𝐴 ⊆ suc 𝐴
26 trssord 4398 . . . 4 ((Tr 𝐴𝐴 ⊆ suc 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
2725, 26mp3an2 1336 . . 3 ((Tr 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
2824, 27mpancom 422 . 2 (Ord suc 𝐴 → Ord 𝐴)
291, 28impbii 126 1 (Ord 𝐴 ↔ Ord suc 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  wal 1362   = wceq 1364  wcel 2160  wss 3144  Tr wtr 4116  Ord word 4380  suc csuc 4383
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-in1 615  ax-in2 616  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  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  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-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4384  df-suc 4389
This theorem is referenced by:  nlimsucg  4583  ordpwsucss  4584
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