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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 8448 | . . . . 5 ⊢ 1o = suc ∅ | |
2 | 1 | difeq2i 4115 | . . . 4 ⊢ (𝐴 ∖ 1o) = (𝐴 ∖ suc ∅) |
3 | 2 | eleq2i 2824 | . . 3 ⊢ (𝐵 ∈ (𝐴 ∖ 1o) ↔ 𝐵 ∈ (𝐴 ∖ suc ∅)) |
4 | eldif 3954 | . . 3 ⊢ (𝐵 ∈ (𝐴 ∖ suc ∅) ↔ (𝐵 ∈ 𝐴 ∧ ¬ 𝐵 ∈ suc ∅)) | |
5 | 3, 4 | bitri 274 | . 2 ⊢ (𝐵 ∈ (𝐴 ∖ 1o) ↔ (𝐵 ∈ 𝐴 ∧ ¬ 𝐵 ∈ suc ∅)) |
6 | 0elon 6407 | . . . . 5 ⊢ ∅ ∈ On | |
7 | ordelord 6375 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | |
8 | ordelsuc 7791 | . . . . 5 ⊢ ((∅ ∈ On ∧ Ord 𝐵) → (∅ ∈ 𝐵 ↔ suc ∅ ⊆ 𝐵)) | |
9 | 6, 7, 8 | sylancr 587 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (∅ ∈ 𝐵 ↔ suc ∅ ⊆ 𝐵)) |
10 | ord0eln0 6408 | . . . . 5 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅)) | |
11 | 7, 10 | syl 17 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅)) |
12 | eloni 6363 | . . . . . 6 ⊢ (∅ ∈ On → Ord ∅) | |
13 | ordsuci 7779 | . . . . . 6 ⊢ (Ord ∅ → Ord suc ∅) | |
14 | 6, 12, 13 | mp2b 10 | . . . . 5 ⊢ Ord suc ∅ |
15 | ordtri1 6386 | . . . . 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 309 | . . 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 282 | 1 ⊢ (Ord 𝐴 → (𝐵 ∈ (𝐴 ∖ 1o) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ≠ wne 2939 ∖ cdif 3941 ⊆ wss 3944 ∅c0 4318 Ord word 6352 Oncon0 6353 suc csuc 6355 1oc1o 8441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-tr 5259 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-ord 6356 df-on 6357 df-suc 6359 df-1o 8448 |
This theorem is referenced by: (None) |
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