Theorem limnsuc 42836

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 42835	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
2		simp3 1135	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) → ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)
3		eqeq1 2729	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 = suc 𝑏 ↔ 𝐴 = suc 𝑏))
4	3	rexbidv 3168	. . . . 5 ⊢ (𝑎 = 𝐴 → (∃𝑏 ∈ On 𝑎 = suc 𝑏 ↔ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
5	4	elrab 3679	. . . 4 ⊢ (𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ↔ (𝐴 ∈ On ∧ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
6	5	simprbi 495	. . 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 216	1 ⊢ (Lim 𝐴 → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∃wrex 3059  {crab 3418  ∅c0 4322  Ord word 6370  Oncon0 6371  Lim wlim 6372  suc csuc 6373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377

This theorem is referenced by: (None)