Theorem onsucf1olem 43795

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 7760	. . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
2	1	3ad2ant1 1142	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∪ 𝐴 ∈ On)
3		eloni 6345	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
4		unizlim 6459	. . . . . . . . . 10 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
5		oran 1000	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) ↔ ¬ (¬ 𝐴 = ∅ ∧ ¬ Lim 𝐴))
6		df-ne 2952	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
7	6	anbi1i 632	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) ↔ (¬ 𝐴 = ∅ ∧ ¬ Lim 𝐴))
8	5, 7	xchbinxr 337	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) ↔ ¬ (𝐴 ≠ ∅ ∧ ¬ Lim 𝐴))
9	4, 8	bitrdi 289	. . . . . . . . 9 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ (𝐴 ≠ ∅ ∧ ¬ Lim 𝐴)))
10	3, 9	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ↔ ¬ (𝐴 ≠ ∅ ∧ ¬ Lim 𝐴)))
11		pm2.21 123	. . . . . . . 8 ⊢ (¬ (𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ((𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → 𝐴 = suc ∪ 𝐴))
12	10, 11	biimtrdi 255	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 → ((𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → 𝐴 = suc ∪ 𝐴)))
13	12	com23 86	. . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)))
14	13	3impib 1125	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
15		idd 24	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = suc ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
16		onuniorsuc 7806	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
17	16	3ad2ant1 1142	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
18	14, 15, 17	mpjaod 869	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → 𝐴 = suc ∪ 𝐴)
19	2, 18	jca 518	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴))
20		eleq1 2844	. . . 4 ⊢ (𝑏 = ∪ 𝐴 → (𝑏 ∈ On ↔ ∪ 𝐴 ∈ On))
21		suceq 6403	. . . . 5 ⊢ (𝑏 = ∪ 𝐴 → suc 𝑏 = suc ∪ 𝐴)
22	21	eqeq2d 2767	. . . 4 ⊢ (𝑏 = ∪ 𝐴 → (𝐴 = suc 𝑏 ↔ 𝐴 = suc ∪ 𝐴))
23	20, 22	anbi12d 640	. . 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 43794	. . 3 ⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
26	25	3ad2ant1 1142	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
27		df-eu 2590	. . 3 ⊢ (∃!𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ↔ (∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ∧ ∃*𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏)))
28		df-reu 3362	. . 3 ⊢ (∃!𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃!𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
29		df-rmo 3361	. . . 4 ⊢ (∃*𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃*𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
30	29	anbi2i 631	. . 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 305	. 2 ⊢ (∃!𝑏 ∈ On 𝐴 = suc 𝑏 ↔ (∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ∧ ∃*𝑏 ∈ On 𝐴 = suc 𝑏))
32	24, 26, 31	sylanbrc 591	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 856   ∧ w3a 1095   = wceq 1554  ∃wex 1793   ∈ wcel 2136  ∃*wmo 2558  ∃!weu 2589   ≠ wne 2951  ∃!wreu 3359  ∃*wrmo 3360  ∅c0 4280  ∪ cuni 4859  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-tr 5202  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341

This theorem is referenced by: (None)