|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > 1ellim | Structured version Visualization version GIF version | ||
| Description: A limit ordinal contains 1. (Contributed by BTernaryTau, 1-Dec-2024.) | 
| Ref | Expression | 
|---|---|
| 1ellim | ⊢ (Lim 𝐴 → 1o ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nlim0 6442 | . . . 4 ⊢ ¬ Lim ∅ | |
| 2 | limeq 6395 | . . . 4 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅)) | |
| 3 | 1, 2 | mtbiri 327 | . . 3 ⊢ (𝐴 = ∅ → ¬ Lim 𝐴) | 
| 4 | 3 | necon2ai 2969 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ ∅) | 
| 5 | nlim1 8528 | . . . 4 ⊢ ¬ Lim 1o | |
| 6 | limeq 6395 | . . . 4 ⊢ (𝐴 = 1o → (Lim 𝐴 ↔ Lim 1o)) | |
| 7 | 5, 6 | mtbiri 327 | . . 3 ⊢ (𝐴 = 1o → ¬ Lim 𝐴) | 
| 8 | 7 | necon2ai 2969 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ 1o) | 
| 9 | limord 6443 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
| 10 | ord1eln01 8535 | . . 3 ⊢ (Ord 𝐴 → (1o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o))) | |
| 11 | 9, 10 | syl 17 | . 2 ⊢ (Lim 𝐴 → (1o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o))) | 
| 12 | 4, 8, 11 | mpbir2and 713 | 1 ⊢ (Lim 𝐴 → 1o ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 Ord word 6382 Lim wlim 6384 1oc1o 8500 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-1o 8507 | 
| This theorem is referenced by: 1onn 8679 | 
| Copyright terms: Public domain | W3C validator |