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Theorem dflim7 43262
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 7868. (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 7868 . 2 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴))
2 ord0eln0 6440 . . . . . 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 1094 . . 3 ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴) ↔ (Ord 𝐴 ∧ (∅ ∈ 𝐴 ∧ ∀𝑏𝐴 suc 𝑏𝐴)))
7 3anass 1094 . . 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 1086  wcel 2105  wne 2937  wral 3058  c0 4338  Ord word 6384  Lim wlim 6386  suc csuc 6387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391
This theorem is referenced by: (None)
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