Theorem limuni3 7828

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 6344	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (Lim 𝑥 ↔ Lim 𝑧))
2	1	rspcv 3584	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Lim 𝑥 → Lim 𝑧))
3		vex 3451	. . . . . . 7 ⊢ 𝑧 ∈ V
4		limelon 6397	. . . . . . 7 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
5	3, 4	mpan 690	. . . . . 6 ⊢ (Lim 𝑧 → 𝑧 ∈ On)
6	2, 5	syl6com 37	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑧 ∈ 𝐴 → 𝑧 ∈ On))
7	6	ssrdv 3952	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → 𝐴 ⊆ On)
8		ssorduni 7755	. . . 4 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
9	7, 8	syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → Ord ∪ 𝐴)
10	9	adantl 481	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → Ord ∪ 𝐴)
11		n0 4316	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
12		0ellim 6396	. . . . . . 7 ⊢ (Lim 𝑧 → ∅ ∈ 𝑧)
13		elunii 4876	. . . . . . . 8 ⊢ ((∅ ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∅ ∈ ∪ 𝐴)
14	13	expcom 413	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (∅ ∈ 𝑧 → ∅ ∈ ∪ 𝐴))
15	12, 14	syl5 34	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (Lim 𝑧 → ∅ ∈ ∪ 𝐴))
16	2, 15	syld 47	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∅ ∈ ∪ 𝐴))
17	16	exlimiv 1930	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∅ ∈ ∪ 𝐴))
18	11, 17	sylbi 217	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∅ ∈ ∪ 𝐴))
19	18	imp 406	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → ∅ ∈ ∪ 𝐴)
20		eluni2 4875	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
21	1	rspccv 3585	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑧 ∈ 𝐴 → Lim 𝑧))
22		limsuc 7825	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → (𝑦 ∈ 𝑧 ↔ suc 𝑦 ∈ 𝑧))
23	22	anbi1d 631	. . . . . . . . . 10 ⊢ (Lim 𝑧 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) ↔ (suc 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴)))
24		elunii 4876	. . . . . . . . . 10 ⊢ ((suc 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → suc 𝑦 ∈ ∪ 𝐴)
25	23, 24	biimtrdi 253	. . . . . . . . 9 ⊢ (Lim 𝑧 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → suc 𝑦 ∈ ∪ 𝐴))
26	25	expd 415	. . . . . . . 8 ⊢ (Lim 𝑧 → (𝑦 ∈ 𝑧 → (𝑧 ∈ 𝐴 → suc 𝑦 ∈ ∪ 𝐴)))
27	26	com3r 87	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (Lim 𝑧 → (𝑦 ∈ 𝑧 → suc 𝑦 ∈ ∪ 𝐴)))
28	21, 27	sylcom 30	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑧 ∈ 𝐴 → (𝑦 ∈ 𝑧 → suc 𝑦 ∈ ∪ 𝐴)))
29	28	rexlimdv 3132	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧 → suc 𝑦 ∈ ∪ 𝐴))
30	20, 29	biimtrid 242	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑦 ∈ ∪ 𝐴 → suc 𝑦 ∈ ∪ 𝐴))
31	30	ralrimiv 3124	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∀𝑦 ∈ ∪ 𝐴 suc 𝑦 ∈ ∪ 𝐴)
32	31	adantl 481	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → ∀𝑦 ∈ ∪ 𝐴 suc 𝑦 ∈ ∪ 𝐴)
33		dflim4 7824	. 2 ⊢ (Lim ∪ 𝐴 ↔ (Ord ∪ 𝐴 ∧ ∅ ∈ ∪ 𝐴 ∧ ∀𝑦 ∈ ∪ 𝐴 suc 𝑦 ∈ ∪ 𝐴))
34	10, 19, 32, 33	syl3anbrc 1344	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → Lim ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  ∪ cuni 4871  Ord word 6331  Oncon0 6332  Lim wlim 6333  suc csuc 6334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338

This theorem is referenced by: (None)