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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ordeldifsucon | Structured version Visualization version GIF version | ||
| Description: Membership in the difference of ordinal and successor ordinal. (Contributed by RP, 16-Jan-2025.) |
| Ref | Expression |
|---|---|
| ordeldifsucon | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eldif 3912 | . 2 ⊢ (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ suc 𝐵)) | |
| 2 | simplr 778 | . . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ On) | |
| 3 | ordelord 6362 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐶 ∈ 𝐴) → Ord 𝐶) | |
| 4 | 3 | adantlr 725 | . . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → Ord 𝐶) |
| 5 | ordelsuc 7794 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ Ord 𝐶) → (𝐵 ∈ 𝐶 ↔ suc 𝐵 ⊆ 𝐶)) | |
| 6 | 2, 4, 5 | syl2anc 593 | . . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → (𝐵 ∈ 𝐶 ↔ suc 𝐵 ⊆ 𝐶)) |
| 7 | eloni 6350 | . . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
| 8 | ordsuci 7785 | . . . . . 6 ⊢ (Ord 𝐵 → Ord suc 𝐵) | |
| 9 | 2, 7, 8 | 3syl 18 | . . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → Ord suc 𝐵) |
| 10 | ordtri1 6373 | . . . . 5 ⊢ ((Ord suc 𝐵 ∧ Ord 𝐶) → (suc 𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ suc 𝐵)) | |
| 11 | 9, 4, 10 | syl2anc 593 | . . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → (suc 𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ suc 𝐵)) |
| 12 | 6, 11 | bitr2d 282 | . . 3 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → (¬ 𝐶 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝐶)) |
| 13 | 12 | pm5.32da 587 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → ((𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶))) |
| 14 | 1, 13 | bitrid 285 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∈ wcel 2141 ∖ cdif 3899 ⊆ wss 3902 Ord word 6339 Oncon0 6340 suc csuc 6342 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-ext 2733 ax-sep 5243 ax-pr 5387 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-sb 2090 df-clab 2740 df-cleq 2753 df-clel 2836 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-tr 5205 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-ord 6343 df-on 6344 df-suc 6346 |
| This theorem is referenced by: orddif0suc 43805 cantnfresb 43861 |
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