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Mirrors > Home > MPE Home > Th. List > ord1eln01 | Structured version Visualization version GIF version |
Description: An ordinal that is not 0 or 1 contains 1. (Contributed by BTernaryTau, 1-Dec-2024.) |
Ref | Expression |
---|---|
ord1eln01 | ⊢ (Ord 𝐴 → (1o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4298 | . . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | 1on 8428 | . . . . . 6 ⊢ 1o ∈ On | |
3 | 2 | onirri 6434 | . . . . 5 ⊢ ¬ 1o ∈ 1o |
4 | eleq2 2823 | . . . . 5 ⊢ (𝐴 = 1o → (1o ∈ 𝐴 ↔ 1o ∈ 1o)) | |
5 | 3, 4 | mtbiri 327 | . . . 4 ⊢ (𝐴 = 1o → ¬ 1o ∈ 𝐴) |
6 | 5 | necon2ai 2970 | . . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ 1o) |
7 | 1, 6 | jca 513 | . 2 ⊢ (1o ∈ 𝐴 → (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o)) |
8 | el1o 8445 | . . . . . . 7 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) | |
9 | 8 | biimpi 215 | . . . . . 6 ⊢ (𝐴 ∈ 1o → 𝐴 = ∅) |
10 | 9 | necon3ai 2965 | . . . . 5 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ∈ 1o) |
11 | nesym 2997 | . . . . . 6 ⊢ (𝐴 ≠ 1o ↔ ¬ 1o = 𝐴) | |
12 | 11 | biimpi 215 | . . . . 5 ⊢ (𝐴 ≠ 1o → ¬ 1o = 𝐴) |
13 | 10, 12 | anim12ci 615 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → (¬ 1o = 𝐴 ∧ ¬ 𝐴 ∈ 1o)) |
14 | pm4.56 988 | . . . 4 ⊢ ((¬ 1o = 𝐴 ∧ ¬ 𝐴 ∈ 1o) ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o)) | |
15 | 13, 14 | sylib 217 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o)) |
16 | 2 | onordi 6432 | . . . 4 ⊢ Ord 1o |
17 | ordtri2 6356 | . . . 4 ⊢ ((Ord 1o ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o))) | |
18 | 16, 17 | mpan 689 | . . 3 ⊢ (Ord 𝐴 → (1o ∈ 𝐴 ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o))) |
19 | 15, 18 | imbitrrid 245 | . 2 ⊢ (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → 1o ∈ 𝐴)) |
20 | 7, 19 | impbid2 225 | 1 ⊢ (Ord 𝐴 → (1o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∅c0 4286 Ord word 6320 1oc1o 8409 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
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 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-tr 5227 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-ord 6324 df-on 6325 df-suc 6327 df-1o 8416 |
This theorem is referenced by: 1ellim 8448 |
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