Theorem onsucf1olem 44025

Description: The successor operation is bijective between the ordinals and the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)

Assertion

Ref	Expression
onsucf1olem	⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏)

Distinct variable group:   𝐴,𝑏

Proof of Theorem onsucf1olem

Step	Hyp	Ref	Expression
1		onuni 7785	. . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
2	1	3ad2ant1 1150	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∪ 𝐴 ∈ On)
3		eloni 6370	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
4		unizlim 6485	. . . . . . . . . 10 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
5		oran 1004	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) ↔ ¬ (¬ 𝐴 = ∅ ∧ ¬ Lim 𝐴))
6		df-ne 2958	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
7	6	anbi1i 635	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) ↔ (¬ 𝐴 = ∅ ∧ ¬ Lim 𝐴))
8	5, 7	xchbinxr 338	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) ↔ ¬ (𝐴 ≠ ∅ ∧ ¬ Lim 𝐴))
9	4, 8	bitrdi 290	. . . . . . . . 9 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ (𝐴 ≠ ∅ ∧ ¬ Lim 𝐴)))
10	3, 9	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ↔ ¬ (𝐴 ≠ ∅ ∧ ¬ Lim 𝐴)))
11		pm2.21 124	. . . . . . . 8 ⊢ (¬ (𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ((𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → 𝐴 = suc ∪ 𝐴))
12	10, 11	biimtrdi 256	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 → ((𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → 𝐴 = suc ∪ 𝐴)))
13	12	com23 87	. . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)))
14	13	3impib 1133	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
15		idd 25	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = suc ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
16		onuniorsuc 7831	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
17	16	3ad2ant1 1150	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
18	14, 15, 17	mpjaod 873	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → 𝐴 = suc ∪ 𝐴)
19	2, 18	jca 520	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴))
20		eleq1 2850	. . . 4 ⊢ (𝑏 = ∪ 𝐴 → (𝑏 ∈ On ↔ ∪ 𝐴 ∈ On))
21		suceq 6429	. . . . 5 ⊢ (𝑏 = ∪ 𝐴 → suc 𝑏 = suc ∪ 𝐴)
22	21	eqeq2d 2773	. . . 4 ⊢ (𝑏 = ∪ 𝐴 → (𝐴 = suc 𝑏 ↔ 𝐴 = suc ∪ 𝐴))
23	20, 22	anbi12d 643	. . 3 ⊢ (𝑏 = ∪ 𝐴 → ((𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ↔ (∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴)))
24	2, 19, 23	spcedv 3556	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
25		onsucf1lem 44024	. . 3 ⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
26	25	3ad2ant1 1150	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
27		df-eu 2596	. . 3 ⊢ (∃!𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ↔ (∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ∧ ∃*𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏)))
28		df-reu 3369	. . 3 ⊢ (∃!𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃!𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
29		df-rmo 3368	. . . 4 ⊢ (∃*𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃*𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
30	29	anbi2i 634	. . 3 ⊢ ((∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ∧ ∃*𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ∧ ∃*𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏)))
31	27, 28, 30	3bitr4i 306	. 2 ⊢ (∃!𝑏 ∈ On 𝐴 = suc 𝑏 ↔ (∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ∧ ∃*𝑏 ∈ On 𝐴 = suc 𝑏))
32	24, 26, 31	sylanbrc 594	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1102   = wceq 1569  ∃wex 1808   ∈ wcel 2142  ∃*wmo 2564  ∃!weu 2595   ≠ wne 2957  ∃!wreu 3366  ∃*wrmo 3367  ∅c0 4285  ∪ cuni 4871  Ord word 6359  Oncon0 6360  Lim wlim 6361  suc csuc 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-tr 5218  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366

This theorem is used by: (None)