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Theorem limuni3 7693
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 6277 . . . . . . 7 (𝑥 = 𝑧 → (Lim 𝑥 ↔ Lim 𝑧))
21rspcv 3556 . . . . . 6 (𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → Lim 𝑧))
3 vex 3435 . . . . . . 7 𝑧 ∈ V
4 limelon 6328 . . . . . . 7 ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
53, 4mpan 687 . . . . . 6 (Lim 𝑧𝑧 ∈ On)
62, 5syl6com 37 . . . . 5 (∀𝑥𝐴 Lim 𝑥 → (𝑧𝐴𝑧 ∈ On))
76ssrdv 3932 . . . 4 (∀𝑥𝐴 Lim 𝑥𝐴 ⊆ On)
8 ssorduni 7623 . . . 4 (𝐴 ⊆ On → Ord 𝐴)
97, 8syl 17 . . 3 (∀𝑥𝐴 Lim 𝑥 → Ord 𝐴)
109adantl 482 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → Ord 𝐴)
11 n0 4286 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧𝐴)
12 0ellim 6327 . . . . . . 7 (Lim 𝑧 → ∅ ∈ 𝑧)
13 elunii 4850 . . . . . . . 8 ((∅ ∈ 𝑧𝑧𝐴) → ∅ ∈ 𝐴)
1413expcom 414 . . . . . . 7 (𝑧𝐴 → (∅ ∈ 𝑧 → ∅ ∈ 𝐴))
1512, 14syl5 34 . . . . . 6 (𝑧𝐴 → (Lim 𝑧 → ∅ ∈ 𝐴))
162, 15syld 47 . . . . 5 (𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1716exlimiv 1937 . . . 4 (∃𝑧 𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1811, 17sylbi 216 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1918imp 407 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → ∅ ∈ 𝐴)
20 eluni2 4849 . . . . 5 (𝑦 𝐴 ↔ ∃𝑧𝐴 𝑦𝑧)
211rspccv 3558 . . . . . . 7 (∀𝑥𝐴 Lim 𝑥 → (𝑧𝐴 → Lim 𝑧))
22 limsuc 7690 . . . . . . . . . . 11 (Lim 𝑧 → (𝑦𝑧 ↔ suc 𝑦𝑧))
2322anbi1d 630 . . . . . . . . . 10 (Lim 𝑧 → ((𝑦𝑧𝑧𝐴) ↔ (suc 𝑦𝑧𝑧𝐴)))
24 elunii 4850 . . . . . . . . . 10 ((suc 𝑦𝑧𝑧𝐴) → suc 𝑦 𝐴)
2523, 24syl6bi 252 . . . . . . . . 9 (Lim 𝑧 → ((𝑦𝑧𝑧𝐴) → suc 𝑦 𝐴))
2625expd 416 . . . . . . . 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 3214 . . . . 5 (∀𝑥𝐴 Lim 𝑥 → (∃𝑧𝐴 𝑦𝑧 → suc 𝑦 𝐴))
3020, 29syl5bi 241 . . . 4 (∀𝑥𝐴 Lim 𝑥 → (𝑦 𝐴 → suc 𝑦 𝐴))
3130ralrimiv 3109 . . 3 (∀𝑥𝐴 Lim 𝑥 → ∀𝑦 𝐴 suc 𝑦 𝐴)
3231adantl 482 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → ∀𝑦 𝐴 suc 𝑦 𝐴)
33 dflim4 7689 . 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 396  wex 1786  wcel 2110  wne 2945  wral 3066  wrex 3067  Vcvv 3431  wss 3892  c0 4262   cuni 4845  Ord word 6264  Oncon0 6265  Lim wlim 6266  suc csuc 6267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5197  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271
This theorem is referenced by: (None)
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