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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 7771 | . . 3 ⊢ (Ord 𝐵 → Ord ∪ 𝐵) | |
2 | ordelord 6377 | . . . 4 ⊢ ((Ord ∪ 𝐵 ∧ 𝐴 ∈ ∪ 𝐵) → Ord 𝐴) | |
3 | 2 | ex 412 | . . 3 ⊢ (Ord ∪ 𝐵 → (𝐴 ∈ ∪ 𝐵 → Ord 𝐴)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 → Ord 𝐴)) |
5 | ordelord 6377 | . . . 4 ⊢ ((Ord 𝐵 ∧ suc 𝐴 ∈ 𝐵) → Ord suc 𝐴) | |
6 | ordsuc 7795 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
7 | 5, 6 | sylibr 233 | . . 3 ⊢ ((Ord 𝐵 ∧ suc 𝐴 ∈ 𝐵) → Ord 𝐴) |
8 | 7 | ex 412 | . 2 ⊢ (Ord 𝐵 → (suc 𝐴 ∈ 𝐵 → Ord 𝐴)) |
9 | ordsson 7764 | . . . . . 6 ⊢ (Ord 𝐵 → 𝐵 ⊆ On) | |
10 | ordunisssuc 6461 | . . . . . 6 ⊢ ((𝐵 ⊆ On ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ suc 𝐴)) | |
11 | 9, 10 | sylan 579 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ suc 𝐴)) |
12 | ordtri1 6388 | . . . . . 6 ⊢ ((Ord ∪ 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ∪ 𝐵)) | |
13 | 1, 12 | sylan 579 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ∪ 𝐵)) |
14 | ordtri1 6388 | . . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴 ∈ 𝐵)) | |
15 | 6, 14 | sylan2b 593 | . . . . 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 412 | . 2 ⊢ (Ord 𝐵 → (Ord 𝐴 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))) |
19 | 4, 8, 18 | pm5.21ndd 379 | 1 ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ⊆ wss 3941 ∪ cuni 4900 Ord word 6354 Oncon0 6355 suc csuc 6357 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
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 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-tr 5257 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-ord 6358 df-on 6359 df-suc 6361 |
This theorem is referenced by: dfac12lem1 10135 dfac12lem2 10136 naddsuc2 42657 |
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