Theorem onsucuni3 38122

Description: If an ordinal number has a predecessor, then it is successor of that predecessor. (Contributed by ML, 17-Oct-2020.)

Assertion

Ref	Expression
onsucuni3	⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc ∪ 𝐵)

Proof of Theorem onsucuni3

Step	Hyp	Ref	Expression
1		eloni 6367	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2	1	3ad2ant1 1151	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → Ord 𝐵)
3		orduniorsuc 7827	. . . 4 ⊢ (Ord 𝐵 → (𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵))
4	2, 3	syl 18	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵))
5	4	orcomd 885	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵))
6		simp2 1155	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 ≠ ∅)
7		df-lim 6362	. . . . . . . 8 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))
8	7	biimpri 231	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵) → Lim 𝐵)
9	8	3expb 1138	. . . . . 6 ⊢ ((Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) → Lim 𝐵)
10	9	con3i 155	. . . . 5 ⊢ (¬ Lim 𝐵 → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)))
11	10	3ad2ant3 1153	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)))
12	2, 11	mpnanrd 415	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))
13	6, 12	mpnanrd 415	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ 𝐵 = ∪ 𝐵)
14		orcom 884	. . 3 ⊢ ((𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵) ↔ (𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵))
15		df-or 862	. . 3 ⊢ ((𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵) ↔ (¬ 𝐵 = ∪ 𝐵 → 𝐵 = suc ∪ 𝐵))
16	14, 15	sylbb 222	. 2 ⊢ ((𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵) → (¬ 𝐵 = ∪ 𝐵 → 𝐵 = suc ∪ 𝐵))
17	5, 13, 16	sylc 66	1 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc ∪ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∅c0 4279  ∪ cuni 4867  Ord word 6356  Oncon0 6357  Lim wlim 6358  suc csuc 6359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363

This theorem is used by:  1oequni2o  38123  rdgsucuni  38124  finxpreclem4  38149