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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 7512 | . . 3 ⊢ (Ord 𝐵 → Ord ∪ 𝐵) | |
2 | ordelord 6216 | . . . 4 ⊢ ((Ord ∪ 𝐵 ∧ 𝐴 ∈ ∪ 𝐵) → Ord 𝐴) | |
3 | 2 | ex 415 | . . 3 ⊢ (Ord ∪ 𝐵 → (𝐴 ∈ ∪ 𝐵 → Ord 𝐴)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 → Ord 𝐴)) |
5 | ordelord 6216 | . . . 4 ⊢ ((Ord 𝐵 ∧ suc 𝐴 ∈ 𝐵) → Ord suc 𝐴) | |
6 | ordsuc 7532 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
7 | 5, 6 | sylibr 236 | . . 3 ⊢ ((Ord 𝐵 ∧ suc 𝐴 ∈ 𝐵) → Ord 𝐴) |
8 | 7 | ex 415 | . 2 ⊢ (Ord 𝐵 → (suc 𝐴 ∈ 𝐵 → Ord 𝐴)) |
9 | ordsson 7507 | . . . . . 6 ⊢ (Ord 𝐵 → 𝐵 ⊆ On) | |
10 | ordunisssuc 6296 | . . . . . 6 ⊢ ((𝐵 ⊆ On ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ suc 𝐴)) | |
11 | 9, 10 | sylan 582 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ suc 𝐴)) |
12 | ordtri1 6227 | . . . . . 6 ⊢ ((Ord ∪ 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ∪ 𝐵)) | |
13 | 1, 12 | sylan 582 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ∪ 𝐵)) |
14 | ordtri1 6227 | . . . . . 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 311 | . . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (¬ 𝐴 ∈ ∪ 𝐵 ↔ ¬ suc 𝐴 ∈ 𝐵)) |
17 | 16 | con4bid 319 | . . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵)) |
18 | 17 | ex 415 | . 2 ⊢ (Ord 𝐵 → (Ord 𝐴 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))) |
19 | 4, 8, 18 | pm5.21ndd 383 | 1 ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 ⊆ wss 3939 ∪ cuni 4841 Ord word 6193 Oncon0 6194 suc csuc 6196 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-tr 5176 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-ord 6197 df-on 6198 df-suc 6200 |
This theorem is referenced by: dfac12lem1 9572 dfac12lem2 9573 |
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