Theorem nlimon 7790

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 6324	. . 3 ⊢ (𝑥 ∈ On → Ord 𝑥)
2		dflim3 7786	. . . . 5 ⊢ (Lim 𝑥 ↔ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
3	2	baib 535	. . . 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 3400	1 ⊢ {𝑥 ∈ On ∣ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)} = {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Syntax hints:  ¬ wn 3   ↔ wb 206   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∃wrex 3057  {crab 3396  ∅c0 4282  Ord word 6313  Oncon0 6314  Lim wlim 6315  suc csuc 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-tr 5203  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320

This theorem is referenced by: (None)