Theorem onsucelab 43295

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 7743	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		eqid 2731	. . 3 ⊢ suc 𝐴 = suc 𝐴
3		id 22	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ On)
4		suceq 6374	. . . . . 6 ⊢ (𝑏 = 𝐴 → suc 𝑏 = suc 𝐴)
5	4	eqeq2d 2742	. . . . 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 3570	. . 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 2735	. . . 4 ⊢ (𝑎 = suc 𝐴 → (𝑎 = suc 𝑏 ↔ suc 𝐴 = suc 𝑏))
10	9	rexbidv 3156	. . 3 ⊢ (𝑎 = suc 𝐴 → (∃𝑏 ∈ On 𝑎 = suc 𝑏 ↔ ∃𝑏 ∈ On suc 𝐴 = suc 𝑏))
11	10	elrab 3647	. 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 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395  Oncon0 6306  suc csuc 6308

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-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

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-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-ord 6309  df-on 6310  df-suc 6312

This theorem is referenced by: (None)