Theorem dflim6 44023

Description: A limit ordinal is a nonzero ordinal which is not a successor ordinal. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)

Assertion

Ref	Expression
dflim6	⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))

Distinct variable group:   𝐴,𝑏

Proof of Theorem dflim6

Step	Hyp	Ref	Expression
1		ioran 999	. . . 4 ⊢ (¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
2		df-ne 2962	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3	2	anbi1i 636	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
4	1, 3	bitr4i 281	. . 3 ⊢ (¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
5	4	anbi2i 635	. 2 ⊢ ((Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏)) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
6		dflim3 7845	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
7		3anass 1111	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
8	5, 6, 7	3bitr4i 306	1 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ≠ wne 2961  ∃wrex 3092  ∅c0 4289  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5260  ax-nul 5272  ax-pr 5407  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-br 5113  df-opab 5177  df-tr 5222  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370

This theorem is used by:  limnsuc  44024