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Theorem ordsuci 7792
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 7795. (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 6375 . . 3 (Ord 𝐴 → Tr 𝐴)
2 suctr 6447 . . 3 (Tr 𝐴 → Tr suc 𝐴)
31, 2syl 17 . 2 (Ord 𝐴 → Tr suc 𝐴)
4 df-suc 6367 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
5 ordsson 7766 . . . 4 (Ord 𝐴𝐴 ⊆ On)
6 elon2 6372 . . . . . 6 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
7 snssi 4810 . . . . . 6 (𝐴 ∈ On → {𝐴} ⊆ On)
86, 7sylbir 234 . . . . 5 ((Ord 𝐴𝐴 ∈ V) → {𝐴} ⊆ On)
9 snprc 4720 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 215 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
11 0ss 4395 . . . . . . 7 ∅ ⊆ On
1210, 11eqsstrdi 4035 . . . . . 6 𝐴 ∈ V → {𝐴} ⊆ On)
1312adantl 482 . . . . 5 ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On)
148, 13pm2.61dan 811 . . . 4 (Ord 𝐴 → {𝐴} ⊆ On)
155, 14unssd 4185 . . 3 (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On)
164, 15eqsstrid 4029 . 2 (Ord 𝐴 → suc 𝐴 ⊆ On)
17 ordon 7760 . . 3 Ord On
1817a1i 11 . 2 (Ord 𝐴 → Ord On)
19 trssord 6378 . 2 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
203, 16, 18, 19syl3anc 1371 1 (Ord 𝐴 → Ord suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cun 3945  wss 3947  c0 4321  {csn 4627  Tr wtr 5264  Ord word 6360  Oncon0 6361  suc csuc 6363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-suc 6367
This theorem is referenced by:  sucexeloni  7793  ordsuc  7797  ord3  8479  ordeldifsucon  41994  ordeldif1o  41995  ordnexbtwnsuc  42002  ordsssucb  42070  onsucunifi  42105
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