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Theorem ordsuci 7817
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 7820. (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 6388 . . 3 (Ord 𝐴 → Tr 𝐴)
2 suctr 6460 . . 3 (Tr 𝐴 → Tr suc 𝐴)
31, 2syl 17 . 2 (Ord 𝐴 → Tr suc 𝐴)
4 df-suc 6380 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
5 ordsson 7791 . . . 4 (Ord 𝐴𝐴 ⊆ On)
6 elon2 6385 . . . . . 6 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
7 snssi 4816 . . . . . 6 (𝐴 ∈ On → {𝐴} ⊆ On)
86, 7sylbir 234 . . . . 5 ((Ord 𝐴𝐴 ∈ V) → {𝐴} ⊆ On)
9 snprc 4726 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 215 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
11 0ss 4400 . . . . . . 7 ∅ ⊆ On
1210, 11eqsstrdi 4036 . . . . . 6 𝐴 ∈ V → {𝐴} ⊆ On)
1312adantl 480 . . . . 5 ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On)
148, 13pm2.61dan 811 . . . 4 (Ord 𝐴 → {𝐴} ⊆ On)
155, 14unssd 4188 . . 3 (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On)
164, 15eqsstrid 4030 . 2 (Ord 𝐴 → suc 𝐴 ⊆ On)
17 ordon 7785 . . 3 Ord On
1817a1i 11 . 2 (Ord 𝐴 → Ord On)
19 trssord 6391 . 2 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
203, 16, 18, 19syl3anc 1368 1 (Ord 𝐴 → Ord suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3473  cun 3947  wss 3949  c0 4326  {csn 4632  Tr wtr 5269  Ord word 6373  Oncon0 6374  suc csuc 6376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378  df-suc 6380
This theorem is referenced by:  sucexeloni  7818  ordsuc  7822  ord3  8510  ordeldifsucon  42719  ordeldif1o  42720  ordnexbtwnsuc  42727  ordsssucb  42795  onsucunifi  42830
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