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Theorem limuni3 7556
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 6196 . . . . . . 7 (𝑥 = 𝑧 → (Lim 𝑥 ↔ Lim 𝑧))
21rspcv 3615 . . . . . 6 (𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → Lim 𝑧))
3 vex 3495 . . . . . . 7 𝑧 ∈ V
4 limelon 6247 . . . . . . 7 ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
53, 4mpan 686 . . . . . 6 (Lim 𝑧𝑧 ∈ On)
62, 5syl6com 37 . . . . 5 (∀𝑥𝐴 Lim 𝑥 → (𝑧𝐴𝑧 ∈ On))
76ssrdv 3970 . . . 4 (∀𝑥𝐴 Lim 𝑥𝐴 ⊆ On)
8 ssorduni 7489 . . . 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 4307 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧𝐴)
12 0ellim 6246 . . . . . . 7 (Lim 𝑧 → ∅ ∈ 𝑧)
13 elunii 4835 . . . . . . . 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 1922 . . . 4 (∃𝑧 𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1811, 17sylbi 218 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1918imp 407 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → ∅ ∈ 𝐴)
20 eluni2 4834 . . . . 5 (𝑦 𝐴 ↔ ∃𝑧𝐴 𝑦𝑧)
211rspccv 3617 . . . . . . 7 (∀𝑥𝐴 Lim 𝑥 → (𝑧𝐴 → Lim 𝑧))
22 limsuc 7553 . . . . . . . . . . 11 (Lim 𝑧 → (𝑦𝑧 ↔ suc 𝑦𝑧))
2322anbi1d 629 . . . . . . . . . 10 (Lim 𝑧 → ((𝑦𝑧𝑧𝐴) ↔ (suc 𝑦𝑧𝑧𝐴)))
24 elunii 4835 . . . . . . . . . 10 ((suc 𝑦𝑧𝑧𝐴) → suc 𝑦 𝐴)
2523, 24syl6bi 254 . . . . . . . . 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 3280 . . . . 5 (∀𝑥𝐴 Lim 𝑥 → (∃𝑧𝐴 𝑦𝑧 → suc 𝑦 𝐴))
3020, 29syl5bi 243 . . . 4 (∀𝑥𝐴 Lim 𝑥 → (𝑦 𝐴 → suc 𝑦 𝐴))
3130ralrimiv 3178 . . 3 (∀𝑥𝐴 Lim 𝑥 → ∀𝑦 𝐴 suc 𝑦 𝐴)
3231adantl 482 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → ∀𝑦 𝐴 suc 𝑦 𝐴)
33 dflim4 7552 . 2 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑦 𝐴 suc 𝑦 𝐴))
3410, 19, 32, 33syl3anbrc 1335 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → Lim 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  wss 3933  c0 4288   cuni 4830  Ord word 6183  Oncon0 6184  Lim wlim 6185  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190
This theorem is referenced by: (None)
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