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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
StepHypRef 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)
54eqeq2d 2748 . . . . 5 (𝑏 = 𝐴 → (suc 𝐴 = suc 𝑏 ↔ suc 𝐴 = suc 𝐴))
65adantl 481 . . . 4 ((𝐴 ∈ On ∧ 𝑏 = 𝐴) → (suc 𝐴 = suc 𝑏 ↔ suc 𝐴 = suc 𝐴))
73, 6rspcedv 3615 . . 3 (𝐴 ∈ On → (suc 𝐴 = suc 𝐴 → ∃𝑏 ∈ On suc 𝐴 = suc 𝑏))
82, 7mpi 20 . 2 (𝐴 ∈ On → ∃𝑏 ∈ On suc 𝐴 = suc 𝑏)
9 eqeq1 2741 . . . 4 (𝑎 = suc 𝐴 → (𝑎 = suc 𝑏 ↔ suc 𝐴 = suc 𝑏))
109rexbidv 3179 . . 3 (𝑎 = suc 𝐴 → (∃𝑏 ∈ On 𝑎 = suc 𝑏 ↔ ∃𝑏 ∈ On suc 𝐴 = suc 𝑏))
1110elrab 3692 . 2 (suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ↔ (suc 𝐴 ∈ On ∧ ∃𝑏 ∈ On suc 𝐴 = suc 𝑏))
121, 8, 11sylanbrc 583 1 (𝐴 ∈ On → suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})
Colors of variables: wff setvar class
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)
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