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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 7807 | . . . 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 6391 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | |
5 | ordsuc 7798 | . . 3 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
6 | ordsuc 7798 | . . 3 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵) | |
7 | ordtri1 6391 | . . 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 2098 ⊆ wss 3943 Ord word 6357 suc csuc 6360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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 6361 df-on 6362 df-suc 6364 |
This theorem is referenced by: oawordri 8551 oeworde 8594 nnawordi 8622 eldifsucnn 8665 ttrcltr 9713 bndrank 9838 rankmapu 9875 ackbij1b 10236 onsuct0 35834 finxpsuclem 36785 onsucwordi 42619 naddgeoa 42726 |
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