Theorem nlimon 7794

Description: Two ways to express the class of non-limit ordinal numbers. Part of Definition 7.27 of [TakeutiZaring] p. 42, who use the symbol K_I for this class. (Contributed by NM, 1-Nov-2004.)

Assertion

Ref	Expression
nlimon	⊢ {𝑥 ∈ On ∣ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)} = {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Distinct variable group:   𝑥,𝑦

Proof of Theorem nlimon

Step	Hyp	Ref	Expression
1		eloni 6323	. . 3 ⊢ (𝑥 ∈ On → Ord 𝑥)
2		dflim3 7790	. . . . 5 ⊢ (Lim 𝑥 ↔ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
3	2	baib 541	. . . 4 ⊢ (Ord 𝑥 → (Lim 𝑥 ↔ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
4	3	con2bid 356	. . 3 ⊢ (Ord 𝑥 → ((𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦) ↔ ¬ Lim 𝑥))
5	1, 4	syl 17	. 2 ⊢ (𝑥 ∈ On → ((𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦) ↔ ¬ Lim 𝑥))
6	5	rabbiia 3397	1 ⊢ {𝑥 ∈ On ∣ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)} = {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Syntax hints:  ¬ wn 3   ↔ wb 208   ∨ wo 854   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  {crab 3393  ∅c0 4263  Ord word 6312  Oncon0 6313  Lim wlim 6314  suc csuc 6315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-tr 5182  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319

This theorem is referenced by: (None)