Theorem dflim6 43297

Description: A limit ordinal is a non-zero 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 985	. . . 4 ⊢ (¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
2		df-ne 2929	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3	2	anbi1i 624	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
4	1, 3	bitr4i 278	. . 3 ⊢ (¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
5	4	anbi2i 623	. 2 ⊢ ((Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏)) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
6		dflim3 7772	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
7		3anass 1094	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
8	5, 6, 7	3bitr4i 303	1 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ≠ wne 2928  ∃wrex 3056  ∅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:  limnsuc  43298