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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordeldif1o | Structured version Visualization version GIF version |
Description: Membership in the difference of ordinal and ordinal one. (Contributed by RP, 16-Jan-2025.) |
Ref | Expression |
---|---|
ordeldif1o | ⊢ (Ord 𝐴 → (𝐵 ∈ (𝐴 ∖ 1o) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 8504 | . . . . 5 ⊢ 1o = suc ∅ | |
2 | 1 | difeq2i 4132 | . . . 4 ⊢ (𝐴 ∖ 1o) = (𝐴 ∖ suc ∅) |
3 | 2 | eleq2i 2830 | . . 3 ⊢ (𝐵 ∈ (𝐴 ∖ 1o) ↔ 𝐵 ∈ (𝐴 ∖ suc ∅)) |
4 | eldif 3972 | . . 3 ⊢ (𝐵 ∈ (𝐴 ∖ suc ∅) ↔ (𝐵 ∈ 𝐴 ∧ ¬ 𝐵 ∈ suc ∅)) | |
5 | 3, 4 | bitri 275 | . 2 ⊢ (𝐵 ∈ (𝐴 ∖ 1o) ↔ (𝐵 ∈ 𝐴 ∧ ¬ 𝐵 ∈ suc ∅)) |
6 | 0elon 6439 | . . . . 5 ⊢ ∅ ∈ On | |
7 | ordelord 6407 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | |
8 | ordelsuc 7839 | . . . . 5 ⊢ ((∅ ∈ On ∧ Ord 𝐵) → (∅ ∈ 𝐵 ↔ suc ∅ ⊆ 𝐵)) | |
9 | 6, 7, 8 | sylancr 587 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (∅ ∈ 𝐵 ↔ suc ∅ ⊆ 𝐵)) |
10 | ord0eln0 6440 | . . . . 5 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅)) | |
11 | 7, 10 | syl 17 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅)) |
12 | eloni 6395 | . . . . . 6 ⊢ (∅ ∈ On → Ord ∅) | |
13 | ordsuci 7827 | . . . . . 6 ⊢ (Ord ∅ → Ord suc ∅) | |
14 | 6, 12, 13 | mp2b 10 | . . . . 5 ⊢ Ord suc ∅ |
15 | ordtri1 6418 | . . . . 5 ⊢ ((Ord suc ∅ ∧ Ord 𝐵) → (suc ∅ ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc ∅)) | |
16 | 14, 7, 15 | sylancr 587 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (suc ∅ ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc ∅)) |
17 | 9, 11, 16 | 3bitr3rd 310 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (¬ 𝐵 ∈ suc ∅ ↔ 𝐵 ≠ ∅)) |
18 | 17 | pm5.32da 579 | . 2 ⊢ (Ord 𝐴 → ((𝐵 ∈ 𝐴 ∧ ¬ 𝐵 ∈ suc ∅) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅))) |
19 | 5, 18 | bitrid 283 | 1 ⊢ (Ord 𝐴 → (𝐵 ∈ (𝐴 ∖ 1o) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 ⊆ wss 3962 ∅c0 4338 Ord word 6384 Oncon0 6385 suc csuc 6387 1oc1o 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 df-suc 6391 df-1o 8504 |
This theorem is referenced by: (None) |
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