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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 5882 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | 1 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → Ord 𝐵) |
3 | orduniorsuc 7183 | . . . 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 402 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵)) |
6 | simp2 1129 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 ≠ ∅) | |
7 | df-lim 5877 | . . . . . . . 8 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) | |
8 | 7 | biimpri 218 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵) → Lim 𝐵) |
9 | 8 | 3expb 1113 | . . . . . 6 ⊢ ((Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) → Lim 𝐵) |
10 | 9 | con3i 150 | . . . . 5 ⊢ (¬ Lim 𝐵 → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))) |
11 | 10 | 3ad2ant3 1127 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))) |
12 | 2, 11 | mpnanrd 33460 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) |
13 | 6, 12 | mpnanrd 33460 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ 𝐵 = ∪ 𝐵) |
14 | orcom 401 | . . 3 ⊢ ((𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵) ↔ (𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵)) | |
15 | df-or 384 | . . 3 ⊢ ((𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵) ↔ (¬ 𝐵 = ∪ 𝐵 → 𝐵 = suc ∪ 𝐵)) | |
16 | 14, 15 | sylbb 209 | . 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 ∨ wo 382 ∧ wa 383 ∧ w3a 1072 = wceq 1620 ∈ wcel 2127 ≠ wne 2920 ∅c0 4046 ∪ cuni 4576 Ord word 5871 Oncon0 5872 Lim wlim 5873 suc csuc 5874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pr 5043 ax-un 7102 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-sbc 3565 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-br 4793 df-opab 4853 df-tr 4893 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 |
This theorem is referenced by: 1oequni2o 33498 rdgsucuni 33499 finxpreclem4 33513 |
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