Theorem onsucf1lem 44054

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 7793	. . 3 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
2		onsucuni2 7836	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝑏) → suc ∪ 𝐴 = 𝐴)
3	2	adantlr 728	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → suc ∪ 𝐴 = 𝐴)
4		simpr 490	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → 𝐴 = suc 𝑏)
5	3, 4	eqtr2d 2801	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → suc 𝑏 = suc ∪ 𝐴)
6	1	anim1i 627	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (∪ 𝐴 ∈ On ∧ 𝑏 ∈ On))
7	6	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → (∪ 𝐴 ∈ On ∧ 𝑏 ∈ On))
8	7	ancomd 467	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → (𝑏 ∈ On ∧ ∪ 𝐴 ∈ On))
9		suc11 6474	. . . . . . 7 ⊢ ((𝑏 ∈ On ∧ ∪ 𝐴 ∈ On) → (suc 𝑏 = suc ∪ 𝐴 ↔ 𝑏 = ∪ 𝐴))
10	8, 9	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → (suc 𝑏 = suc ∪ 𝐴 ↔ 𝑏 = ∪ 𝐴))
11	5, 10	mpbid 235	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → 𝑏 = ∪ 𝐴)
12	11	ex 418	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 = suc 𝑏 → 𝑏 = ∪ 𝐴))
13	12	ralrimiva 3159	. . 3 ⊢ (𝐴 ∈ On → ∀𝑏 ∈ On (𝐴 = suc 𝑏 → 𝑏 = ∪ 𝐴))
14		eqeq2 2777	. . . . 5 ⊢ (𝑐 = ∪ 𝐴 → (𝑏 = 𝑐 ↔ 𝑏 = ∪ 𝐴))
15	14	imbi2d 343	. . . 4 ⊢ (𝑐 = ∪ 𝐴 → ((𝐴 = suc 𝑏 → 𝑏 = 𝑐) ↔ (𝐴 = suc 𝑏 → 𝑏 = ∪ 𝐴)))
16	15	ralbidv 3190	. . 3 ⊢ (𝑐 = ∪ 𝐴 → (∀𝑏 ∈ On (𝐴 = suc 𝑏 → 𝑏 = 𝑐) ↔ ∀𝑏 ∈ On (𝐴 = suc 𝑏 → 𝑏 = ∪ 𝐴)))
17	1, 13, 16	spcedv 3559	. 2 ⊢ (𝐴 ∈ On → ∃𝑐∀𝑏 ∈ On (𝐴 = suc 𝑏 → 𝑏 = 𝑐))
18		nfv 1947	. . 3 ⊢ Ⅎ𝑐 𝐴 = suc 𝑏
19	18	rmo2 3841	. 2 ⊢ (∃*𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃𝑐∀𝑏 ∈ On (𝐴 = suc 𝑏 → 𝑏 = 𝑐))
20	17, 19	sylibr 237	1 ⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3081  ∃*wrmo 3370  ∪ cuni 4874  Oncon0 6364  suc csuc 6366

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-un 7742

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 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-on 6368  df-suc 6370

This theorem is used by:  onsucf1olem  44055