Theorem onsucuni3 37380

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 6312	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2	1	3ad2ant1 1133	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → Ord 𝐵)
3		orduniorsuc 7755	. . . 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 871	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵))
6		simp2 1137	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 ≠ ∅)
7		df-lim 6307	. . . . . . . 8 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))
8	7	biimpri 228	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵) → Lim 𝐵)
9	8	3expb 1120	. . . . . 6 ⊢ ((Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) → Lim 𝐵)
10	9	con3i 154	. . . . 5 ⊢ (¬ Lim 𝐵 → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)))
11	10	3ad2ant3 1135	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)))
12	2, 11	mpnanrd 409	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))
13	6, 12	mpnanrd 409	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ 𝐵 = ∪ 𝐵)
14		orcom 870	. . 3 ⊢ ((𝐵 = suc ∪ 𝐵 ∨ 𝐵 = ∪ 𝐵) ↔ (𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵))
15		df-or 848	. . 3 ⊢ ((𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵) ↔ (¬ 𝐵 = ∪ 𝐵 → 𝐵 = suc ∪ 𝐵))
16	14, 15	sylbb 219	. 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)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110   ≠ wne 2926  ∅c0 4281  ∪ cuni 4857  Ord word 6301  Oncon0 6302  Lim wlim 6303  suc csuc 6304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-tr 5197  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308

This theorem is referenced by:  1oequni2o  37381  rdgsucuni  37382  finxpreclem4  37407