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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 4338 | . . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | 1on 8507 | . . . . . 6 ⊢ 1o ∈ On | |
3 | 2 | onirri 6487 | . . . . 5 ⊢ ¬ 1o ∈ 1o |
4 | eleq2 2818 | . . . . 5 ⊢ (𝐴 = 1o → (1o ∈ 𝐴 ↔ 1o ∈ 1o)) | |
5 | 3, 4 | mtbiri 326 | . . . 4 ⊢ (𝐴 = 1o → ¬ 1o ∈ 𝐴) |
6 | 5 | necon2ai 2967 | . . 3 ⊢ (1o ∈ 𝐴 → 𝐴 ≠ 1o) |
7 | 1, 6 | jca 510 | . 2 ⊢ (1o ∈ 𝐴 → (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o)) |
8 | el1o 8524 | . . . . . . 7 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) | |
9 | 8 | biimpi 215 | . . . . . 6 ⊢ (𝐴 ∈ 1o → 𝐴 = ∅) |
10 | 9 | necon3ai 2962 | . . . . 5 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ∈ 1o) |
11 | nesym 2994 | . . . . . 6 ⊢ (𝐴 ≠ 1o ↔ ¬ 1o = 𝐴) | |
12 | 11 | biimpi 215 | . . . . 5 ⊢ (𝐴 ≠ 1o → ¬ 1o = 𝐴) |
13 | 10, 12 | anim12ci 612 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → (¬ 1o = 𝐴 ∧ ¬ 𝐴 ∈ 1o)) |
14 | pm4.56 986 | . . . 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 6485 | . . . 4 ⊢ Ord 1o |
17 | ordtri2 6409 | . . . 4 ⊢ ((Ord 1o ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ ¬ (1o = 𝐴 ∨ 𝐴 ∈ 1o))) | |
18 | 16, 17 | mpan 688 | . . 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 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∅c0 4326 Ord word 6373 1oc1o 8488 |
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 df-1o 8495 |
This theorem is referenced by: 1ellim 8527 |
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