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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 7795. (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 6375 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | suctr 6447 | . . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (Ord 𝐴 → Tr suc 𝐴) |
4 | df-suc 6367 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | ordsson 7766 | . . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
6 | elon2 6372 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | |
7 | snssi 4810 | . . . . . 6 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On) | |
8 | 6, 7 | sylbir 234 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → {𝐴} ⊆ On) |
9 | snprc 4720 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | 9 | biimpi 215 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
11 | 0ss 4395 | . . . . . . 7 ⊢ ∅ ⊆ On | |
12 | 10, 11 | eqsstrdi 4035 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} ⊆ On) |
13 | 12 | adantl 482 | . . . . 5 ⊢ ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On) |
14 | 8, 13 | pm2.61dan 811 | . . . 4 ⊢ (Ord 𝐴 → {𝐴} ⊆ On) |
15 | 5, 14 | unssd 4185 | . . 3 ⊢ (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On) |
16 | 4, 15 | eqsstrid 4029 | . 2 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ On) |
17 | ordon 7760 | . . 3 ⊢ Ord On | |
18 | 17 | a1i 11 | . 2 ⊢ (Ord 𝐴 → Ord On) |
19 | trssord 6378 | . 2 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴) | |
20 | 3, 16, 18, 19 | syl3anc 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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