Theorem limuni3 7844

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 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		limeq 6370	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (Lim 𝑥 ↔ Lim 𝑧))
2	1	rspcv 3586	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Lim 𝑥 → Lim 𝑧))
3		vex 3467	. . . . . . 7 ⊢ 𝑧 ∈ V
4		limelon 6424	. . . . . . 7 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
5	3, 4	mpan 702	. . . . . 6 ⊢ (Lim 𝑧 → 𝑧 ∈ On)
6	2, 5	syl6com 38	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑧 ∈ 𝐴 → 𝑧 ∈ On))
7	6	ssrdv 3951	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → 𝐴 ⊆ On)
8		ssorduni 7774	. . . 4 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
9	7, 8	syl 18	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → Ord ∪ 𝐴)
10	9	adantl 486	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → Ord ∪ 𝐴)
11		n0 4314	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
12		0ellim 6423	. . . . . . 7 ⊢ (Lim 𝑧 → ∅ ∈ 𝑧)
13		elunii 4878	. . . . . . . 8 ⊢ ((∅ ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∅ ∈ ∪ 𝐴)
14	13	expcom 418	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (∅ ∈ 𝑧 → ∅ ∈ ∪ 𝐴))
15	12, 14	syl5 35	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (Lim 𝑧 → ∅ ∈ ∪ 𝐴))
16	2, 15	syld 48	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∅ ∈ ∪ 𝐴))
17	16	exlimiv 1957	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∅ ∈ ∪ 𝐴))
18	11, 17	sylbi 220	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∅ ∈ ∪ 𝐴))
19	18	imp 411	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → ∅ ∈ ∪ 𝐴)
20		eluni2 4877	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
21	1	rspccv 3587	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑧 ∈ 𝐴 → Lim 𝑧))
22		limsuc 7841	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → (𝑦 ∈ 𝑧 ↔ suc 𝑦 ∈ 𝑧))
23	22	anbi1d 642	. . . . . . . . . 10 ⊢ (Lim 𝑧 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) ↔ (suc 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴)))
24		elunii 4878	. . . . . . . . . 10 ⊢ ((suc 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → suc 𝑦 ∈ ∪ 𝐴)
25	23, 24	biimtrdi 256	. . . . . . . . 9 ⊢ (Lim 𝑧 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → suc 𝑦 ∈ ∪ 𝐴))
26	25	expd 420	. . . . . . . 8 ⊢ (Lim 𝑧 → (𝑦 ∈ 𝑧 → (𝑧 ∈ 𝐴 → suc 𝑦 ∈ ∪ 𝐴)))
27	26	com3r 88	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (Lim 𝑧 → (𝑦 ∈ 𝑧 → suc 𝑦 ∈ ∪ 𝐴)))
28	21, 27	sylcom 31	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑧 ∈ 𝐴 → (𝑦 ∈ 𝑧 → suc 𝑦 ∈ ∪ 𝐴)))
29	28	rexlimdv 3170	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧 → suc 𝑦 ∈ ∪ 𝐴))
30	20, 29	biimtrid 245	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑦 ∈ ∪ 𝐴 → suc 𝑦 ∈ ∪ 𝐴))
31	30	ralrimiv 3162	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∀𝑦 ∈ ∪ 𝐴 suc 𝑦 ∈ ∪ 𝐴)
32	31	adantl 486	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → ∀𝑦 ∈ ∪ 𝐴 suc 𝑦 ∈ ∪ 𝐴)
33		dflim4 7840	. 2 ⊢ (Lim ∪ 𝐴 ↔ (Ord ∪ 𝐴 ∧ ∅ ∈ ∪ 𝐴 ∧ ∀𝑦 ∈ ∪ 𝐴 suc 𝑦 ∈ ∪ 𝐴))
34	10, 19, 32, 33	syl3anbrc 1360	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → Lim ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ∪ cuni 4873  Ord word 6357  Oncon0 6358  Lim wlim 6359  suc csuc 6360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364

This theorem is referenced by: (None)