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Theorem ordsuci 7828
Description: The successor of an ordinal class is an ordinal class. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 6-Jun-1994.) Extract and adapt from a subproof of onsuc 7831. (Revised by BTernaryTau, 6-Jan-2025.) (Proof shortened by BJ, 11-Jan-2025.)
Assertion
Ref Expression
ordsuci (Ord 𝐴 → Ord suc 𝐴)

Proof of Theorem ordsuci
StepHypRef Expression
1 ordtr 6398 . . 3 (Ord 𝐴 → Tr 𝐴)
2 suctr 6470 . . 3 (Tr 𝐴 → Tr suc 𝐴)
31, 2syl 17 . 2 (Ord 𝐴 → Tr suc 𝐴)
4 df-suc 6390 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
5 ordsson 7803 . . . 4 (Ord 𝐴𝐴 ⊆ On)
6 elon2 6395 . . . . . 6 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
7 snssi 4808 . . . . . 6 (𝐴 ∈ On → {𝐴} ⊆ On)
86, 7sylbir 235 . . . . 5 ((Ord 𝐴𝐴 ∈ V) → {𝐴} ⊆ On)
9 snprc 4717 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 216 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
11 0ss 4400 . . . . . . 7 ∅ ⊆ On
1210, 11eqsstrdi 4028 . . . . . 6 𝐴 ∈ V → {𝐴} ⊆ On)
1312adantl 481 . . . . 5 ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On)
148, 13pm2.61dan 813 . . . 4 (Ord 𝐴 → {𝐴} ⊆ On)
155, 14unssd 4192 . . 3 (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On)
164, 15eqsstrid 4022 . 2 (Ord 𝐴 → suc 𝐴 ⊆ On)
17 ordon 7797 . . 3 Ord On
1817a1i 11 . 2 (Ord 𝐴 → Ord On)
19 trssord 6401 . 2 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
203, 16, 18, 19syl3anc 1373 1 (Ord 𝐴 → Ord suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  wss 3951  c0 4333  {csn 4626  Tr wtr 5259  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  sucexeloni  7829  ordsuc  7833  ord3  8523  ordeldifsucon  43272  ordeldif1o  43273  ordnexbtwnsuc  43280  ordsssucb  43348  onsucunifi  43383
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