Theorem limuni3 7674

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 6263	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (Lim 𝑥 ↔ Lim 𝑧))
2	1	rspcv 3547	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Lim 𝑥 → Lim 𝑧))
3		vex 3426	. . . . . . 7 ⊢ 𝑧 ∈ V
4		limelon 6314	. . . . . . 7 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
5	3, 4	mpan 686	. . . . . 6 ⊢ (Lim 𝑧 → 𝑧 ∈ On)
6	2, 5	syl6com 37	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑧 ∈ 𝐴 → 𝑧 ∈ On))
7	6	ssrdv 3923	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → 𝐴 ⊆ On)
8		ssorduni 7606	. . . 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 4277	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
12		0ellim 6313	. . . . . . 7 ⊢ (Lim 𝑧 → ∅ ∈ 𝑧)
13		elunii 4841	. . . . . . . 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 1934	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∅ ∈ ∪ 𝐴))
18	11, 17	sylbi 216	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∅ ∈ ∪ 𝐴))
19	18	imp 406	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → ∅ ∈ ∪ 𝐴)
20		eluni2 4840	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
21	1	rspccv 3549	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑧 ∈ 𝐴 → Lim 𝑧))
22		limsuc 7671	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → (𝑦 ∈ 𝑧 ↔ suc 𝑦 ∈ 𝑧))
23	22	anbi1d 629	. . . . . . . . . 10 ⊢ (Lim 𝑧 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) ↔ (suc 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴)))
24		elunii 4841	. . . . . . . . . 10 ⊢ ((suc 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → suc 𝑦 ∈ ∪ 𝐴)
25	23, 24	syl6bi 252	. . . . . . . . 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 3211	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧 → suc 𝑦 ∈ ∪ 𝐴))
30	20, 29	syl5bi 241	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑦 ∈ ∪ 𝐴 → suc 𝑦 ∈ ∪ 𝐴))
31	30	ralrimiv 3106	. . 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 7670	. 2 ⊢ (Lim ∪ 𝐴 ↔ (Ord ∪ 𝐴 ∧ ∅ ∈ ∪ 𝐴 ∧ ∀𝑦 ∈ ∪ 𝐴 suc 𝑦 ∈ ∪ 𝐴))
34	10, 19, 32, 33	syl3anbrc 1341	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → Lim ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  Ord word 6250  Oncon0 6251  Lim wlim 6252  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257

This theorem is referenced by: (None)