Theorem nlimon 7776

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 6311	. . 3 ⊢ (𝑥 ∈ On → Ord 𝑥)
2		dflim3 7772	. . . . 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 3399	1 ⊢ {𝑥 ∈ On ∣ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)} = {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Syntax hints:  ¬ wn 3   ↔ wb 206   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395  ∅c0 4278  Ord word 6300  Oncon0 6301  Lim wlim 6302  suc csuc 6303

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307

This theorem is referenced by: (None)