Theorem dflim3 7892

Description: An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (Contributed by NM, 1-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dflim3	⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem dflim3

Step	Hyp	Ref	Expression
1		df-lim 6393	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
2		3anass 1111	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)))
3		df-ne 2960	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
4	3	a1i 11	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅))
5		orduninsuc 7888	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
6	4, 5	anbi12d 644	. . . 4 ⊢ (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
7		ioran 999	. . . 4 ⊢ (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
8	6, 7	bitr4di 292	. . 3 ⊢ (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
9	8	pm5.32i 585	. 2 ⊢ ((Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
10	1, 2, 9	3bitri 300	1 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ≠ wne 2959  ∃wrex 3090  ∅c0 4281  ∪ cuni 4872  Ord word 6387  Oncon0 6388  Lim wlim 6389  suc csuc 6390

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394

This theorem is used by:  nlimon  7896  tfinds  7905  oalimcl  8606  omlimcl  8624  r1wunlim  10861  dflim6  44527  naddgeoa  44657  faosnf0.11b  44689  dfsucon  44785