Theorem onsucf1olem 42509

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 7769	. . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
2	1	3ad2ant1 1130	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∪ 𝐴 ∈ On)
3		eloni 6364	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
4		unizlim 6477	. . . . . . . . . 10 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
5		oran 986	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) ↔ ¬ (¬ 𝐴 = ∅ ∧ ¬ Lim 𝐴))
6		df-ne 2933	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
7	6	anbi1i 623	. . . . . . . . . . 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	syl6bi 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 1113	. . . . 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 7818	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
17	16	3ad2ant1 1130	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
18	14, 15, 17	mpjaod 857	. . . 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 2813	. . . 4 ⊢ (𝑏 = ∪ 𝐴 → (𝑏 ∈ On ↔ ∪ 𝐴 ∈ On))
21		suceq 6420	. . . . 5 ⊢ (𝑏 = ∪ 𝐴 → suc 𝑏 = suc ∪ 𝐴)
22	21	eqeq2d 2735	. . . 4 ⊢ (𝑏 = ∪ 𝐴 → (𝐴 = suc 𝑏 ↔ 𝐴 = suc ∪ 𝐴))
23	20, 22	anbi12d 630	. . 3 ⊢ (𝑏 = ∪ 𝐴 → ((𝑏 ∈ On ∧ 𝐴 = suc 𝑏) ↔ (∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴)))
24	2, 19, 23	spcedv 3580	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃𝑏(𝑏 ∈ On ∧ 𝐴 = suc 𝑏))
25		onsucf1lem 42508	. . 3 ⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
26	25	3ad2ant1 1130	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
27		df-eu 2555	. . 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 622	. . 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 582	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃*wmo 2524  ∃!weu 2554   ≠ wne 2932  ∃!wreu 3366  ∃*wrmo 3367  ∅c0 4314  ∪ cuni 4899  Ord word 6353  Oncon0 6354  Lim wlim 6355  suc csuc 6356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360

This theorem is referenced by: (None)