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Theorem limuni3 7872
Description: The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.)
Assertion
Ref Expression
limuni3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → Lim 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem limuni3
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limeq 6397 . . . . . . 7 (𝑥 = 𝑧 → (Lim 𝑥 ↔ Lim 𝑧))
21rspcv 3617 . . . . . 6 (𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → Lim 𝑧))
3 vex 3481 . . . . . . 7 𝑧 ∈ V
4 limelon 6449 . . . . . . 7 ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
53, 4mpan 690 . . . . . 6 (Lim 𝑧𝑧 ∈ On)
62, 5syl6com 37 . . . . 5 (∀𝑥𝐴 Lim 𝑥 → (𝑧𝐴𝑧 ∈ On))
76ssrdv 4000 . . . 4 (∀𝑥𝐴 Lim 𝑥𝐴 ⊆ On)
8 ssorduni 7797 . . . 4 (𝐴 ⊆ On → Ord 𝐴)
97, 8syl 17 . . 3 (∀𝑥𝐴 Lim 𝑥 → Ord 𝐴)
109adantl 481 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → Ord 𝐴)
11 n0 4358 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧𝐴)
12 0ellim 6448 . . . . . . 7 (Lim 𝑧 → ∅ ∈ 𝑧)
13 elunii 4916 . . . . . . . 8 ((∅ ∈ 𝑧𝑧𝐴) → ∅ ∈ 𝐴)
1413expcom 413 . . . . . . 7 (𝑧𝐴 → (∅ ∈ 𝑧 → ∅ ∈ 𝐴))
1512, 14syl5 34 . . . . . 6 (𝑧𝐴 → (Lim 𝑧 → ∅ ∈ 𝐴))
162, 15syld 47 . . . . 5 (𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1716exlimiv 1927 . . . 4 (∃𝑧 𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1811, 17sylbi 217 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1918imp 406 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → ∅ ∈ 𝐴)
20 eluni2 4915 . . . . 5 (𝑦 𝐴 ↔ ∃𝑧𝐴 𝑦𝑧)
211rspccv 3618 . . . . . . 7 (∀𝑥𝐴 Lim 𝑥 → (𝑧𝐴 → Lim 𝑧))
22 limsuc 7869 . . . . . . . . . . 11 (Lim 𝑧 → (𝑦𝑧 ↔ suc 𝑦𝑧))
2322anbi1d 631 . . . . . . . . . 10 (Lim 𝑧 → ((𝑦𝑧𝑧𝐴) ↔ (suc 𝑦𝑧𝑧𝐴)))
24 elunii 4916 . . . . . . . . . 10 ((suc 𝑦𝑧𝑧𝐴) → suc 𝑦 𝐴)
2523, 24biimtrdi 253 . . . . . . . . 9 (Lim 𝑧 → ((𝑦𝑧𝑧𝐴) → suc 𝑦 𝐴))
2625expd 415 . . . . . . . 8 (Lim 𝑧 → (𝑦𝑧 → (𝑧𝐴 → suc 𝑦 𝐴)))
2726com3r 87 . . . . . . 7 (𝑧𝐴 → (Lim 𝑧 → (𝑦𝑧 → suc 𝑦 𝐴)))
2821, 27sylcom 30 . . . . . 6 (∀𝑥𝐴 Lim 𝑥 → (𝑧𝐴 → (𝑦𝑧 → suc 𝑦 𝐴)))
2928rexlimdv 3150 . . . . 5 (∀𝑥𝐴 Lim 𝑥 → (∃𝑧𝐴 𝑦𝑧 → suc 𝑦 𝐴))
3020, 29biimtrid 242 . . . 4 (∀𝑥𝐴 Lim 𝑥 → (𝑦 𝐴 → suc 𝑦 𝐴))
3130ralrimiv 3142 . . 3 (∀𝑥𝐴 Lim 𝑥 → ∀𝑦 𝐴 suc 𝑦 𝐴)
3231adantl 481 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → ∀𝑦 𝐴 suc 𝑦 𝐴)
33 dflim4 7868 . 2 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑦 𝐴 suc 𝑦 𝐴))
3410, 19, 32, 33syl3anbrc 1342 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → Lim 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1775  wcel 2105  wne 2937  wral 3058  wrex 3067  Vcvv 3477  wss 3962  c0 4338   cuni 4911  Ord word 6384  Oncon0 6385  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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