Theorem dflim6 43882

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 2965	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3	2	anbi1i 635	. . . 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 634	. 2 ⊢ ((Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏)) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
6		dflim3 7842	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
7		3anass 1109	. 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))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ≠ wne 2964  ∃wrex 3095  ∅c0 4294  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 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367

This theorem is referenced by:  limnsuc  43883