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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 8506 | . . . . 5 ⊢ 1o = suc ∅ | |
| 2 | 1 | difeq2i 4123 | . . . 4 ⊢ (𝐴 ∖ 1o) = (𝐴 ∖ suc ∅) |
| 3 | 2 | eleq2i 2833 | . . 3 ⊢ (𝐵 ∈ (𝐴 ∖ 1o) ↔ 𝐵 ∈ (𝐴 ∖ suc ∅)) |
| 4 | eldif 3961 | . . 3 ⊢ (𝐵 ∈ (𝐴 ∖ suc ∅) ↔ (𝐵 ∈ 𝐴 ∧ ¬ 𝐵 ∈ suc ∅)) | |
| 5 | 3, 4 | bitri 275 | . 2 ⊢ (𝐵 ∈ (𝐴 ∖ 1o) ↔ (𝐵 ∈ 𝐴 ∧ ¬ 𝐵 ∈ suc ∅)) |
| 6 | 0elon 6438 | . . . . 5 ⊢ ∅ ∈ On | |
| 7 | ordelord 6406 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | |
| 8 | ordelsuc 7840 | . . . . 5 ⊢ ((∅ ∈ On ∧ Ord 𝐵) → (∅ ∈ 𝐵 ↔ suc ∅ ⊆ 𝐵)) | |
| 9 | 6, 7, 8 | sylancr 587 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (∅ ∈ 𝐵 ↔ suc ∅ ⊆ 𝐵)) |
| 10 | ord0eln0 6439 | . . . . 5 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅)) | |
| 11 | 7, 10 | syl 17 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅)) |
| 12 | eloni 6394 | . . . . . 6 ⊢ (∅ ∈ On → Ord ∅) | |
| 13 | ordsuci 7828 | . . . . . 6 ⊢ (Ord ∅ → Ord suc ∅) | |
| 14 | 6, 12, 13 | mp2b 10 | . . . . 5 ⊢ Ord suc ∅ |
| 15 | ordtri1 6417 | . . . . 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 2108 ≠ wne 2940 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 Ord word 6383 Oncon0 6384 suc csuc 6386 1oc1o 8499 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 df-1o 8506 |
| This theorem is referenced by: (None) |
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