Theorem onsucf1olem 43368

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 7727	. . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
2	1	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∪ 𝐴 ∈ On)
3		eloni 6322	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
4		unizlim 6436	. . . . . . . . . 10 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
5		oran 991	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) ↔ ¬ (¬ 𝐴 = ∅ ∧ ¬ Lim 𝐴))
6		df-ne 2929	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
7	6	anbi1i 624	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) ↔ (¬ 𝐴 = ∅ ∧ ¬ Lim 𝐴))
8	5, 7	xchbinxr 335	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) ↔ ¬ (𝐴 ≠ ∅ ∧ ¬ Lim 𝐴))
9	4, 8	bitrdi 287	. . . . . . . . 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 253	. . . . . . 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 1116	. . . . 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 7773	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
17	16	3ad2ant1 1133	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
18	14, 15, 17	mpjaod 860	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → 𝐴 = suc ∪ 𝐴)
19	2, 18	jca 511	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴))
20		eleq1 2819	. . . 4 ⊢ (𝑏 = ∪ 𝐴 → (𝑏 ∈ On ↔ ∪ 𝐴 ∈ On))
21		suceq 6380	. . . . 5 ⊢ (𝑏 = ∪ 𝐴 → suc 𝑏 = suc ∪ 𝐴)
22	21	eqeq2d 2742	. . . 4 ⊢ (𝑏 = ∪ 𝐴 → (𝐴 = suc 𝑏 ↔ 𝐴 = suc ∪ 𝐴))
23	20, 22	anbi12d 632	. . 3 ⊢ (𝑏 = ∪ 𝐴 → ((𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ↔ (∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴)))
24	2, 19, 23	spcedv 3548	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
25		onsucf1lem 43367	. . 3 ⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
26	25	3ad2ant1 1133	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
27		df-eu 2564	. . 3 ⊢ (∃!𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ↔ (∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ∧ ∃*𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏)))
28		df-reu 3347	. . 3 ⊢ (∃!𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃!𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
29		df-rmo 3346	. . . 4 ⊢ (∃*𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃*𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
30	29	anbi2i 623	. . 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 303	. 2 ⊢ (∃!𝑏 ∈ On 𝐴 = suc 𝑏 ↔ (∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ∧ ∃*𝑏 ∈ On 𝐴 = suc 𝑏))
32	24, 26, 31	sylanbrc 583	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃*wmo 2533  ∃!weu 2563   ≠ wne 2928  ∃!wreu 3344  ∃*wrmo 3345  ∅c0 4282  ∪ cuni 4858  Ord word 6311  Oncon0 6312  Lim wlim 6313  suc csuc 6314

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

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-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-tr 5201  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318

This theorem is referenced by: (None)