Theorem onsucf1olem 44215

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 1151	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∪ 𝐴 ∈ On)
3		eloni 6361	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
4		unizlim 6476	. . . . . . . . . 10 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
5		oran 1005	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) ↔ ¬ (¬ 𝐴 = ∅ ∧ ¬ Lim 𝐴))
6		df-ne 2956	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
7	6	anbi1i 636	. . . . . . . . . . 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 1134	. . . . 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 1151	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
18	14, 15, 17	mpjaod 874	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → 𝐴 = suc ∪ 𝐴)
19	2, 18	jca 521	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴))
20		eleq1 2848	. . . 4 ⊢ (𝑏 = ∪ 𝐴 → (𝑏 ∈ On ↔ ∪ 𝐴 ∈ On))
21		suceq 6420	. . . . 5 ⊢ (𝑏 = ∪ 𝐴 → suc 𝑏 = suc ∪ 𝐴)
22	21	eqeq2d 2771	. . . 4 ⊢ (𝑏 = ∪ 𝐴 → (𝐴 = suc 𝑏 ↔ 𝐴 = suc ∪ 𝐴))
23	20, 22	anbi12d 644	. . 3 ⊢ (𝑏 = ∪ 𝐴 → ((𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ↔ (∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴)))
24	2, 19, 23	spcedv 3552	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
25		onsucf1lem 44214	. . 3 ⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
26	25	3ad2ant1 1151	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
27		df-eu 2594	. . 3 ⊢ (∃!𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ↔ (∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ∧ ∃*𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏)))
28		df-reu 3366	. . 3 ⊢ (∃!𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃!𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
29		df-rmo 3365	. . . 4 ⊢ (∃*𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃*𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
30	29	anbi2i 635	. . 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 595	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2562  ∃!weu 2593   ≠ wne 2955  ∃!wreu 3363  ∃*wrmo 3364  ∅c0 4278  ∪ cuni 4866  Ord word 6350  Oncon0 6351  Lim wlim 6352  suc csuc 6353

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7734

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-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357

This theorem is used by: (None)