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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsucuni3 | Structured version Visualization version GIF version |
Description: If an ordinal number has a predecessor, then it is successor of that predecessor. (Contributed by ML, 17-Oct-2020.) |
Ref | Expression |
---|---|
onsucuni3 | ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6386 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | 1 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → Ord 𝐵) |
3 | orduniorsuc 7839 | . . . 4 ⊢ (Ord 𝐵 → (𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵)) |
5 | 4 | orcomd 869 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵)) |
6 | simp2 1134 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 ≠ ∅) | |
7 | df-lim 6381 | . . . . . . . 8 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) | |
8 | 7 | biimpri 227 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵) → Lim 𝐵) |
9 | 8 | 3expb 1117 | . . . . . 6 ⊢ ((Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) → Lim 𝐵) |
10 | 9 | con3i 154 | . . . . 5 ⊢ (¬ Lim 𝐵 → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))) |
11 | 10 | 3ad2ant3 1132 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))) |
12 | 2, 11 | mpnanrd 408 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) |
13 | 6, 12 | mpnanrd 408 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ 𝐵 = ∪ 𝐵) |
14 | orcom 868 | . . 3 ⊢ ((𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵) ↔ (𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵)) | |
15 | df-or 846 | . . 3 ⊢ ((𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵) ↔ (¬ 𝐵 = ∪ 𝐵 → 𝐵 = suc ∪ 𝐵)) | |
16 | 14, 15 | sylbb 218 | . 2 ⊢ ((𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵) → (¬ 𝐵 = ∪ 𝐵 → 𝐵 = suc ∪ 𝐵)) |
17 | 5, 13, 16 | sylc 65 | 1 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∅c0 4325 ∪ cuni 4913 Ord word 6375 Oncon0 6376 Lim wlim 6377 suc csuc 6378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-tr 5271 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 |
This theorem is referenced by: 1oequni2o 37075 rdgsucuni 37076 finxpreclem4 37101 |
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