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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 6400 . . 3 (Ord 𝐴 → Tr 𝐴)
2 suctr 6472 . . 3 (Tr 𝐴 → Tr suc 𝐴)
31, 2syl 17 . 2 (Ord 𝐴 → Tr suc 𝐴)
4 df-suc 6392 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
5 ordsson 7802 . . . 4 (Ord 𝐴𝐴 ⊆ On)
6 elon2 6397 . . . . . 6 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
7 snssi 4813 . . . . . 6 (𝐴 ∈ On → {𝐴} ⊆ On)
86, 7sylbir 235 . . . . 5 ((Ord 𝐴𝐴 ∈ V) → {𝐴} ⊆ On)
9 snprc 4722 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 216 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
11 0ss 4406 . . . . . . 7 ∅ ⊆ On
1210, 11eqsstrdi 4050 . . . . . 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 4202 . . 3 (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On)
164, 15eqsstrid 4044 . 2 (Ord 𝐴 → suc 𝐴 ⊆ On)
17 ordon 7796 . . 3 Ord On
1817a1i 11 . 2 (Ord 𝐴 → Ord On)
19 trssord 6403 . 2 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
203, 16, 18, 19syl3anc 1370 1 (Ord 𝐴 → Ord suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  wss 3963  c0 4339  {csn 4631  Tr wtr 5265  Ord word 6385  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  sucexeloni  7829  ordsuc  7833  ord3  8522  ordeldifsucon  43249  ordeldif1o  43250  ordnexbtwnsuc  43257  ordsssucb  43325  onsucunifi  43360
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