|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 7831. (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 6398 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
| 2 | suctr 6470 | . . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (Ord 𝐴 → Tr suc 𝐴) | 
| 4 | df-suc 6390 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 5 | ordsson 7803 | . . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
| 6 | elon2 6395 | . . . . . 6 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | |
| 7 | snssi 4808 | . . . . . 6 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On) | |
| 8 | 6, 7 | sylbir 235 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → {𝐴} ⊆ On) | 
| 9 | snprc 4717 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 10 | 9 | biimpi 216 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) | 
| 11 | 0ss 4400 | . . . . . . 7 ⊢ ∅ ⊆ On | |
| 12 | 10, 11 | eqsstrdi 4028 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} ⊆ On) | 
| 13 | 12 | adantl 481 | . . . . 5 ⊢ ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On) | 
| 14 | 8, 13 | pm2.61dan 813 | . . . 4 ⊢ (Ord 𝐴 → {𝐴} ⊆ On) | 
| 15 | 5, 14 | unssd 4192 | . . 3 ⊢ (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On) | 
| 16 | 4, 15 | eqsstrid 4022 | . 2 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ On) | 
| 17 | ordon 7797 | . . 3 ⊢ Ord On | |
| 18 | 17 | a1i 11 | . 2 ⊢ (Ord 𝐴 → Ord On) | 
| 19 | trssord 6401 | . 2 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴) | |
| 20 | 3, 16, 18, 19 | syl3anc 1373 | 1 ⊢ (Ord 𝐴 → Ord suc 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 {csn 4626 Tr wtr 5259 Ord word 6383 Oncon0 6384 suc csuc 6386 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 | 
| This theorem is referenced by: sucexeloni 7829 ordsuc 7833 ord3 8523 ordeldifsucon 43272 ordeldif1o 43273 ordnexbtwnsuc 43280 ordsssucb 43348 onsucunifi 43383 | 
| Copyright terms: Public domain | W3C validator |