Theorem onsucelab 43691

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 7764	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		eqid 2736	. . 3 ⊢ suc 𝐴 = suc 𝐴
3		id 22	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ On)
4		suceq 6391	. . . . . 6 ⊢ (𝑏 = 𝐴 → suc 𝑏 = suc 𝐴)
5	4	eqeq2d 2747	. . . . 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 3557	. . 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 2740	. . . 4 ⊢ (𝑎 = suc 𝐴 → (𝑎 = suc 𝑏 ↔ suc 𝐴 = suc 𝑏))
10	9	rexbidv 3161	. . 3 ⊢ (𝑎 = suc 𝐴 → (∃𝑏 ∈ On 𝑎 = suc 𝑏 ↔ ∃𝑏 ∈ On suc 𝐴 = suc 𝑏))
11	10	elrab 3634	. 2 ⊢ (suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ↔ (suc 𝐴 ∈ On ∧ ∃𝑏 ∈ On suc 𝐴 = suc 𝑏))
12	1, 8, 11	sylanbrc 584	1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  {crab 3389  Oncon0 6323  suc csuc 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-suc 6329

This theorem is referenced by: (None)