Theorem dflim3 7827

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 6351	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
2		3anass 1106	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)))
3		df-ne 2958	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
4	3	a1i 11	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅))
5		orduninsuc 7823	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
6	4, 5	anbi12d 641	. . . 4 ⊢ (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
7		ioran 997	. . . 4 ⊢ (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
8	6, 7	bitr4di 291	. . 3 ⊢ (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
9	8	pm5.32i 582	. 2 ⊢ ((Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
10	1, 2, 9	3bitri 299	1 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560   ≠ wne 2957  ∃wrex 3086  ∅c0 4285  ∪ cuni 4865  Ord word 6345  Oncon0 6346  Lim wlim 6347  suc csuc 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352

This theorem is referenced by:  nlimon  7831  tfinds  7840  oalimcl  8529  omlimcl  8547  r1wunlim  10695  dflim6  43841  naddgeoa  43971  faosnf0.11b  44003  dfsucon  44099