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Theorem ordsuci 7748
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 7751. (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 6336 . . 3 (Ord 𝐴 → Tr 𝐴)
2 suctr 6408 . . 3 (Tr 𝐴 → Tr suc 𝐴)
31, 2syl 17 . 2 (Ord 𝐴 → Tr suc 𝐴)
4 df-suc 6328 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
5 ordsson 7722 . . . 4 (Ord 𝐴𝐴 ⊆ On)
6 elon2 6333 . . . . . 6 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
7 snssi 4773 . . . . . 6 (𝐴 ∈ On → {𝐴} ⊆ On)
86, 7sylbir 234 . . . . 5 ((Ord 𝐴𝐴 ∈ V) → {𝐴} ⊆ On)
9 snprc 4683 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 215 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
11 0ss 4361 . . . . . . 7 ∅ ⊆ On
1210, 11eqsstrdi 4003 . . . . . 6 𝐴 ∈ V → {𝐴} ⊆ On)
1312adantl 483 . . . . 5 ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On)
148, 13pm2.61dan 812 . . . 4 (Ord 𝐴 → {𝐴} ⊆ On)
155, 14unssd 4151 . . 3 (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On)
164, 15eqsstrid 3997 . 2 (Ord 𝐴 → suc 𝐴 ⊆ On)
17 ordon 7716 . . 3 Ord On
1817a1i 11 . 2 (Ord 𝐴 → Ord On)
19 trssord 6339 . 2 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
203, 16, 18, 19syl3anc 1372 1 (Ord 𝐴 → Ord suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3448  cun 3913  wss 3915  c0 4287  {csn 4591  Tr wtr 5227  Ord word 6321  Oncon0 6322  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  sucexeloni  7749  ordsuc  7753  ord3  8434  ordeldifsucon  41623  ordeldif1o  41624  ordnexbtwnsuc  41631  onsucunifi  41715
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