Theorem onsucelab 43276

Description: The successor of every ordinal is an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)

Assertion

Ref	Expression
onsucelab	⊢ (𝐴 ∈ On → suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})

Distinct variable group:   𝐴,𝑎,𝑏

Proof of Theorem onsucelab

Step	Hyp	Ref	Expression
1		onsuc 7831	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		eqid 2737	. . 3 ⊢ suc 𝐴 = suc 𝐴
3		id 22	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ On)
4		suceq 6450	. . . . . 6 ⊢ (𝑏 = 𝐴 → suc 𝑏 = suc 𝐴)
5	4	eqeq2d 2748	. . . . 5 ⊢ (𝑏 = 𝐴 → (suc 𝐴 = suc 𝑏 ↔ suc 𝐴 = suc 𝐴))
6	5	adantl 481	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑏 = 𝐴) → (suc 𝐴 = suc 𝑏 ↔ suc 𝐴 = suc 𝐴))
7	3, 6	rspcedv 3615	. . 3 ⊢ (𝐴 ∈ On → (suc 𝐴 = suc 𝐴 → ∃𝑏 ∈ On suc 𝐴 = suc 𝑏))
8	2, 7	mpi 20	. 2 ⊢ (𝐴 ∈ On → ∃𝑏 ∈ On suc 𝐴 = suc 𝑏)
9		eqeq1 2741	. . . 4 ⊢ (𝑎 = suc 𝐴 → (𝑎 = suc 𝑏 ↔ suc 𝐴 = suc 𝑏))
10	9	rexbidv 3179	. . 3 ⊢ (𝑎 = suc 𝐴 → (∃𝑏 ∈ On 𝑎 = suc 𝑏 ↔ ∃𝑏 ∈ On suc 𝐴 = suc 𝑏))
11	10	elrab 3692	. 2 ⊢ (suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ↔ (suc 𝐴 ∈ On ∧ ∃𝑏 ∈ On suc 𝐴 = suc 𝑏))
12	1, 8, 11	sylanbrc 583	1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  {crab 3436  Oncon0 6384  suc csuc 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390

This theorem is referenced by: (None)