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Theorem limuni3 7631
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 6225 . . . . . . 7 (𝑥 = 𝑧 → (Lim 𝑥 ↔ Lim 𝑧))
21rspcv 3532 . . . . . 6 (𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → Lim 𝑧))
3 vex 3412 . . . . . . 7 𝑧 ∈ V
4 limelon 6276 . . . . . . 7 ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
53, 4mpan 690 . . . . . 6 (Lim 𝑧𝑧 ∈ On)
62, 5syl6com 37 . . . . 5 (∀𝑥𝐴 Lim 𝑥 → (𝑧𝐴𝑧 ∈ On))
76ssrdv 3907 . . . 4 (∀𝑥𝐴 Lim 𝑥𝐴 ⊆ On)
8 ssorduni 7563 . . . 4 (𝐴 ⊆ On → Ord 𝐴)
97, 8syl 17 . . 3 (∀𝑥𝐴 Lim 𝑥 → Ord 𝐴)
109adantl 485 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → Ord 𝐴)
11 n0 4261 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧𝐴)
12 0ellim 6275 . . . . . . 7 (Lim 𝑧 → ∅ ∈ 𝑧)
13 elunii 4824 . . . . . . . 8 ((∅ ∈ 𝑧𝑧𝐴) → ∅ ∈ 𝐴)
1413expcom 417 . . . . . . 7 (𝑧𝐴 → (∅ ∈ 𝑧 → ∅ ∈ 𝐴))
1512, 14syl5 34 . . . . . 6 (𝑧𝐴 → (Lim 𝑧 → ∅ ∈ 𝐴))
162, 15syld 47 . . . . 5 (𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1716exlimiv 1938 . . . 4 (∃𝑧 𝑧𝐴 → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1811, 17sylbi 220 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 Lim 𝑥 → ∅ ∈ 𝐴))
1918imp 410 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → ∅ ∈ 𝐴)
20 eluni2 4823 . . . . 5 (𝑦 𝐴 ↔ ∃𝑧𝐴 𝑦𝑧)
211rspccv 3534 . . . . . . 7 (∀𝑥𝐴 Lim 𝑥 → (𝑧𝐴 → Lim 𝑧))
22 limsuc 7628 . . . . . . . . . . 11 (Lim 𝑧 → (𝑦𝑧 ↔ suc 𝑦𝑧))
2322anbi1d 633 . . . . . . . . . 10 (Lim 𝑧 → ((𝑦𝑧𝑧𝐴) ↔ (suc 𝑦𝑧𝑧𝐴)))
24 elunii 4824 . . . . . . . . . 10 ((suc 𝑦𝑧𝑧𝐴) → suc 𝑦 𝐴)
2523, 24syl6bi 256 . . . . . . . . 9 (Lim 𝑧 → ((𝑦𝑧𝑧𝐴) → suc 𝑦 𝐴))
2625expd 419 . . . . . . . 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 3202 . . . . 5 (∀𝑥𝐴 Lim 𝑥 → (∃𝑧𝐴 𝑦𝑧 → suc 𝑦 𝐴))
3020, 29syl5bi 245 . . . 4 (∀𝑥𝐴 Lim 𝑥 → (𝑦 𝐴 → suc 𝑦 𝐴))
3130ralrimiv 3104 . . 3 (∀𝑥𝐴 Lim 𝑥 → ∀𝑦 𝐴 suc 𝑦 𝐴)
3231adantl 485 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → ∀𝑦 𝐴 suc 𝑦 𝐴)
33 dflim4 7627 . 2 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑦 𝐴 suc 𝑦 𝐴))
3410, 19, 32, 33syl3anbrc 1345 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → Lim 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wex 1787  wcel 2110  wne 2940  wral 3061  wrex 3062  Vcvv 3408  wss 3866  c0 4237   cuni 4819  Ord word 6212  Oncon0 6213  Lim wlim 6214  suc csuc 6215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219
This theorem is referenced by: (None)
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