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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 6079 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | 1 | 3ad2ant1 1126 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → Ord 𝐵) |
3 | orduniorsuc 7404 | . . . 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 866 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵)) |
6 | simp2 1130 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 ≠ ∅) | |
7 | df-lim 6074 | . . . . . . . 8 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) | |
8 | 7 | biimpri 229 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵) → Lim 𝐵) |
9 | 8 | 3expb 1113 | . . . . . 6 ⊢ ((Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) → Lim 𝐵) |
10 | 9 | con3i 157 | . . . . 5 ⊢ (¬ Lim 𝐵 → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))) |
11 | 10 | 3ad2ant3 1128 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))) |
12 | 2, 11 | mpnanrd 410 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) |
13 | 6, 12 | mpnanrd 410 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ 𝐵 = ∪ 𝐵) |
14 | orcom 865 | . . 3 ⊢ ((𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵) ↔ (𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵)) | |
15 | df-or 843 | . . 3 ⊢ ((𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵) ↔ (¬ 𝐵 = ∪ 𝐵 → 𝐵 = suc ∪ 𝐵)) | |
16 | 14, 15 | sylbb 220 | . 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 396 ∨ wo 842 ∧ w3a 1080 = wceq 1522 ∈ wcel 2080 ≠ wne 2983 ∅c0 4213 ∪ cuni 4747 Ord word 6068 Oncon0 6069 Lim wlim 6070 suc csuc 6071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pr 5224 ax-un 7322 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-br 4965 df-opab 5027 df-tr 5067 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 |
This theorem is referenced by: 1oequni2o 34193 rdgsucuni 34194 finxpreclem4 34219 |
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