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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 7814 | . . . 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 6397 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | |
5 | ordsuc 7805 | . . 3 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
6 | ordsuc 7805 | . . 3 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵) | |
7 | ordtri1 6397 | . . 3 ⊢ ((Ord suc 𝐴 ∧ Ord suc 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴)) | |
8 | 5, 6, 7 | syl2anb 597 | . 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 205 ∧ wa 395 ∈ wcel 2105 ⊆ wss 3948 Ord word 6363 suc csuc 6366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 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 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-ord 6367 df-on 6368 df-suc 6370 |
This theorem is referenced by: oawordri 8556 oeworde 8599 nnawordi 8627 eldifsucnn 8669 ttrcltr 9717 bndrank 9842 rankmapu 9879 ackbij1b 10240 onsuct0 35792 finxpsuclem 36744 onsucwordi 42503 naddgeoa 42610 |
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