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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 7820. (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 6388 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | suctr 6460 | . . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (Ord 𝐴 → Tr suc 𝐴) |
4 | df-suc 6380 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | ordsson 7791 | . . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
6 | elon2 6385 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | |
7 | snssi 4816 | . . . . . 6 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On) | |
8 | 6, 7 | sylbir 234 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → {𝐴} ⊆ On) |
9 | snprc 4726 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | 9 | biimpi 215 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
11 | 0ss 4400 | . . . . . . 7 ⊢ ∅ ⊆ On | |
12 | 10, 11 | eqsstrdi 4036 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} ⊆ On) |
13 | 12 | adantl 480 | . . . . 5 ⊢ ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On) |
14 | 8, 13 | pm2.61dan 811 | . . . 4 ⊢ (Ord 𝐴 → {𝐴} ⊆ On) |
15 | 5, 14 | unssd 4188 | . . 3 ⊢ (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On) |
16 | 4, 15 | eqsstrid 4030 | . 2 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ On) |
17 | ordon 7785 | . . 3 ⊢ Ord On | |
18 | 17 | a1i 11 | . 2 ⊢ (Ord 𝐴 → Ord On) |
19 | trssord 6391 | . 2 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴) | |
20 | 3, 16, 18, 19 | syl3anc 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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