Theorem limnsuc 43790

Description: A limit ordinal is not an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)

Assertion

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

Distinct variable group:   𝐴,𝑎,𝑏

Proof of Theorem limnsuc

Step	Hyp	Ref	Expression
1		dflim6 43789	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
2		simp3 1147	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) → ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)
3		eqeq1 2760	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 = suc 𝑏 ↔ 𝐴 = suc 𝑏))
4	3	rexbidv 3180	. . . . 5 ⊢ (𝑎 = 𝐴 → (∃𝑏 ∈ On 𝑎 = suc 𝑏 ↔ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
5	4	elrab 3645	. . . 4 ⊢ (𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ↔ (𝐴 ∈ On ∧ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
6	5	simprbi 500	. . 3 ⊢ (𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} → ∃𝑏 ∈ On 𝐴 = suc 𝑏)
7	2, 6	nsyl 140	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})
8	1, 7	sylbi 219	1 ⊢ (Lim 𝐴 → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136   ≠ wne 2951  ∃wrex 3080  {crab 3408  ∅c0 4280  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-tr 5202  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341

This theorem is referenced by: (None)