Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dflim7 Structured version   Visualization version   GIF version

Theorem dflim7 43286
Description: A limit ordinal is a non-zero ordinal that contains all the successors of its elements. Lemma 1.18 of [Schloeder] p. 2. Closely related to dflim4 7869. (Contributed by RP, 17-Jan-2025.)
Assertion
Ref Expression
dflim7 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴𝐴 ≠ ∅))
Distinct variable group:   𝐴,𝑏

Proof of Theorem dflim7
StepHypRef Expression
1 dflim4 7869 . 2 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴))
2 ord0eln0 6439 . . . . . 6 (Ord 𝐴 → (∅ ∈ 𝐴𝐴 ≠ ∅))
32anbi1d 631 . . . . 5 (Ord 𝐴 → ((∅ ∈ 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴) ↔ (𝐴 ≠ ∅ ∧ ∀𝑏𝐴 suc 𝑏𝐴)))
43biancomd 463 . . . 4 (Ord 𝐴 → ((∅ ∈ 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴) ↔ (∀𝑏𝐴 suc 𝑏𝐴𝐴 ≠ ∅)))
54pm5.32i 574 . . 3 ((Ord 𝐴 ∧ (∅ ∈ 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴)) ↔ (Ord 𝐴 ∧ (∀𝑏𝐴 suc 𝑏𝐴𝐴 ≠ ∅)))
6 3anass 1095 . . 3 ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴) ↔ (Ord 𝐴 ∧ (∅ ∈ 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴)))
7 3anass 1095 . . 3 ((Ord 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴𝐴 ≠ ∅) ↔ (Ord 𝐴 ∧ (∀𝑏𝐴 suc 𝑏𝐴𝐴 ≠ ∅)))
85, 6, 73bitr4i 303 . 2 ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴) ↔ (Ord 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴𝐴 ≠ ∅))
91, 8bitri 275 1 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087  wcel 2108  wne 2940  wral 3061  c0 4333  Ord word 6383  Lim wlim 6385  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator