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Mirrors > Home > MPE Home > Th. List > ordsuci | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ordsuci | ⊢ (Ord 𝐴 → Ord suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 6336 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | suctr 6408 | . . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (Ord 𝐴 → Tr suc 𝐴) |
4 | df-suc 6328 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | ordsson 7722 | . . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
6 | elon2 6333 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | |
7 | snssi 4773 | . . . . . 6 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On) | |
8 | 6, 7 | sylbir 234 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → {𝐴} ⊆ On) |
9 | snprc 4683 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | 9 | biimpi 215 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
11 | 0ss 4361 | . . . . . . 7 ⊢ ∅ ⊆ On | |
12 | 10, 11 | eqsstrdi 4003 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} ⊆ On) |
13 | 12 | adantl 483 | . . . . 5 ⊢ ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On) |
14 | 8, 13 | pm2.61dan 812 | . . . 4 ⊢ (Ord 𝐴 → {𝐴} ⊆ On) |
15 | 5, 14 | unssd 4151 | . . 3 ⊢ (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On) |
16 | 4, 15 | eqsstrid 3997 | . 2 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ On) |
17 | ordon 7716 | . . 3 ⊢ Ord On | |
18 | 17 | a1i 11 | . 2 ⊢ (Ord 𝐴 → Ord On) |
19 | trssord 6339 | . 2 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴) | |
20 | 3, 16, 18, 19 | syl3anc 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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