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| Mirrors > Home > MPE Home > Th. List > ordsucsssuc | Structured version Visualization version GIF version | ||
| Description: The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.) |
| Ref | Expression |
|---|---|
| ordsucsssuc | ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ordsucelsuc 7842 | . . . 4 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ suc 𝐴)) | |
| 2 | 1 | notbid 318 | . . 3 ⊢ (Ord 𝐴 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ suc 𝐵 ∈ suc 𝐴)) |
| 3 | 2 | adantr 480 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵 ∈ 𝐴 ↔ ¬ suc 𝐵 ∈ suc 𝐴)) |
| 4 | ordtri1 6417 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | |
| 5 | ordsuc 7833 | . . 3 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
| 6 | ordsuc 7833 | . . 3 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵) | |
| 7 | ordtri1 6417 | . . 3 ⊢ ((Ord suc 𝐴 ∧ Ord suc 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴)) | |
| 8 | 5, 6, 7 | syl2anb 598 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴)) |
| 9 | 3, 4, 8 | 3bitr4d 311 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 Ord word 6383 suc csuc 6386 |
| 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 ax-un 7755 |
| 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 |
| This theorem is referenced by: oawordri 8588 oeworde 8631 nnawordi 8659 eldifsucnn 8702 ttrcltr 9756 bndrank 9881 rankmapu 9918 ackbij1b 10278 pw2bday 28418 onsuct0 36442 finxpsuclem 37398 onsucwordi 43301 naddgeoa 43407 |
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