Theorem onsucf1lem 43453

Description: For ordinals, the successor operation is injective, so there is at most one ordinal that a given ordinal could be the successor of. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)

Assertion

Ref	Expression
onsucf1lem	⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)

Distinct variable group:   𝐴,𝑏

Proof of Theorem onsucf1lem

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onuni 7731	. . 3 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
2		onsucuni2 7774	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝑏) → suc ∪ 𝐴 = 𝐴)
3	2	adantlr 715	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → suc ∪ 𝐴 = 𝐴)
4		simpr 484	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → 𝐴 = suc 𝑏)
5	3, 4	eqtr2d 2770	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → suc 𝑏 = suc ∪ 𝐴)
6	1	anim1i 615	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (∪ 𝐴 ∈ On ∧ 𝑏 ∈ On))
7	6	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → (∪ 𝐴 ∈ On ∧ 𝑏 ∈ On))
8	7	ancomd 461	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → (𝑏 ∈ On ∧ ∪ 𝐴 ∈ On))
9		suc11 6424	. . . . . . 7 ⊢ ((𝑏 ∈ On ∧ ∪ 𝐴 ∈ On) → (suc 𝑏 = suc ∪ 𝐴 ↔ 𝑏 = ∪ 𝐴))
10	8, 9	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → (suc 𝑏 = suc ∪ 𝐴 ↔ 𝑏 = ∪ 𝐴))
11	5, 10	mpbid 232	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → 𝑏 = ∪ 𝐴)
12	11	ex 412	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 = suc 𝑏 → 𝑏 = ∪ 𝐴))
13	12	ralrimiva 3126	. . 3 ⊢ (𝐴 ∈ On → ∀𝑏 ∈ On (𝐴 = suc 𝑏 → 𝑏 = ∪ 𝐴))
14		eqeq2 2746	. . . . 5 ⊢ (𝑐 = ∪ 𝐴 → (𝑏 = 𝑐 ↔ 𝑏 = ∪ 𝐴))
15	14	imbi2d 340	. . . 4 ⊢ (𝑐 = ∪ 𝐴 → ((𝐴 = suc 𝑏 → 𝑏 = 𝑐) ↔ (𝐴 = suc 𝑏 → 𝑏 = ∪ 𝐴)))
16	15	ralbidv 3157	. . 3 ⊢ (𝑐 = ∪ 𝐴 → (∀𝑏 ∈ On (𝐴 = suc 𝑏 → 𝑏 = 𝑐) ↔ ∀𝑏 ∈ On (𝐴 = suc 𝑏 → 𝑏 = ∪ 𝐴)))
17	1, 13, 16	spcedv 3550	. 2 ⊢ (𝐴 ∈ On → ∃𝑐∀𝑏 ∈ On (𝐴 = suc 𝑏 → 𝑏 = 𝑐))
18		nfv 1915	. . 3 ⊢ Ⅎ𝑐 𝐴 = suc 𝑏
19	18	rmo2 3835	. 2 ⊢ (∃*𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃𝑐∀𝑏 ∈ On (𝐴 = suc 𝑏 → 𝑏 = 𝑐))
20	17, 19	sylibr 234	1 ⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3049  ∃*wrmo 3347  ∪ cuni 4861  Oncon0 6315  suc csuc 6317

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

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 2537  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-on 6319  df-suc 6321

This theorem is referenced by:  onsucf1olem  43454