Theorem nlimon 7836

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 6371	. . 3 ⊢ (𝑥 ∈ On → Ord 𝑥)
2		dflim3 7832	. . . . 5 ⊢ (Lim 𝑥 ↔ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
3	2	baib 536	. . . 4 ⊢ (Ord 𝑥 → (Lim 𝑥 ↔ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
4	3	con2bid 354	. . 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 3436	1 ⊢ {𝑥 ∈ On ∣ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)} = {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Syntax hints:  ¬ wn 3   ↔ wb 205   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  {crab 3432  ∅c0 4321  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367

This theorem is referenced by: (None)