Theorem limnsuc 43256

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 43255	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
2		simp3 1138	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) → ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)
3		eqeq1 2740	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 = suc 𝑏 ↔ 𝐴 = suc 𝑏))
4	3	rexbidv 3165	. . . . 5 ⊢ (𝑎 = 𝐴 → (∃𝑏 ∈ On 𝑎 = suc 𝑏 ↔ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
5	4	elrab 3676	. . . 4 ⊢ (𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ↔ (𝐴 ∈ On ∧ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
6	5	simprbi 496	. . 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 217	1 ⊢ (Lim 𝐴 → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061  {crab 3420  ∅c0 4313  Ord word 6356  Oncon0 6357  Lim wlim 6358  suc csuc 6359

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363

This theorem is referenced by: (None)