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Mirrors > Home > MPE Home > Th. List > ordsucuniel | Structured version Visualization version GIF version |
Description: Given an element 𝐴 of the union of an ordinal 𝐵, suc 𝐴 is an element of 𝐵 itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.) |
Ref | Expression |
---|---|
ordsucuniel | ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orduni 7729 | . . 3 ⊢ (Ord 𝐵 → Ord ∪ 𝐵) | |
2 | ordelord 6344 | . . . 4 ⊢ ((Ord ∪ 𝐵 ∧ 𝐴 ∈ ∪ 𝐵) → Ord 𝐴) | |
3 | 2 | ex 414 | . . 3 ⊢ (Ord ∪ 𝐵 → (𝐴 ∈ ∪ 𝐵 → Ord 𝐴)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 → Ord 𝐴)) |
5 | ordelord 6344 | . . . 4 ⊢ ((Ord 𝐵 ∧ suc 𝐴 ∈ 𝐵) → Ord suc 𝐴) | |
6 | ordsuc 7753 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
7 | 5, 6 | sylibr 233 | . . 3 ⊢ ((Ord 𝐵 ∧ suc 𝐴 ∈ 𝐵) → Ord 𝐴) |
8 | 7 | ex 414 | . 2 ⊢ (Ord 𝐵 → (suc 𝐴 ∈ 𝐵 → Ord 𝐴)) |
9 | ordsson 7722 | . . . . . 6 ⊢ (Ord 𝐵 → 𝐵 ⊆ On) | |
10 | ordunisssuc 6428 | . . . . . 6 ⊢ ((𝐵 ⊆ On ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ suc 𝐴)) | |
11 | 9, 10 | sylan 581 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ suc 𝐴)) |
12 | ordtri1 6355 | . . . . . 6 ⊢ ((Ord ∪ 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ∪ 𝐵)) | |
13 | 1, 12 | sylan 581 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ∪ 𝐵)) |
14 | ordtri1 6355 | . . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴 ∈ 𝐵)) | |
15 | 6, 14 | sylan2b 595 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴 ∈ 𝐵)) |
16 | 11, 13, 15 | 3bitr3d 309 | . . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (¬ 𝐴 ∈ ∪ 𝐵 ↔ ¬ suc 𝐴 ∈ 𝐵)) |
17 | 16 | con4bid 317 | . . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵)) |
18 | 17 | ex 414 | . 2 ⊢ (Ord 𝐵 → (Ord 𝐴 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))) |
19 | 4, 8, 18 | pm5.21ndd 381 | 1 ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3915 ∪ cuni 4870 Ord word 6321 Oncon0 6322 suc csuc 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 df-suc 6328 |
This theorem is referenced by: dfac12lem1 10086 dfac12lem2 10087 naddsuc2 41738 |
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