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 ≠ ∅ ∧ ∀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 6225	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (Lim 𝑥 ↔ Lim 𝑧))
2	1	rspcv 3532	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Lim 𝑥 → Lim 𝑧))
3		vex 3412	. . . . . . 7 ⊢ 𝑧 ∈ V
4		limelon 6276	. . . . . . 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 3907	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → 𝐴 ⊆ On)
8		ssorduni 7563	. . . 4 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
9	7, 8	syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → Ord ∪ 𝐴)
10	9	adantl 485	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → Ord ∪ 𝐴)
11		n0 4261	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
12		0ellim 6275	. . . . . . 7 ⊢ (Lim 𝑧 → ∅ ∈ 𝑧)
13		elunii 4824	. . . . . . . 8 ⊢ ((∅ ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∅ ∈ ∪ 𝐴)
14	13	expcom 417	. . . . . . 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 1938	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∅ ∈ ∪ 𝐴))
18	11, 17	sylbi 220	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∅ ∈ ∪ 𝐴))
19	18	imp 410	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → ∅ ∈ ∪ 𝐴)
20		eluni2 4823	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
21	1	rspccv 3534	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑧 ∈ 𝐴 → Lim 𝑧))
22		limsuc 7628	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → (𝑦 ∈ 𝑧 ↔ suc 𝑦 ∈ 𝑧))
23	22	anbi1d 633	. . . . . . . . . 10 ⊢ (Lim 𝑧 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) ↔ (suc 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴)))
24		elunii 4824	. . . . . . . . . 10 ⊢ ((suc 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → suc 𝑦 ∈ ∪ 𝐴)
25	23, 24	syl6bi 256	. . . . . . . . 9 ⊢ (Lim 𝑧 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → suc 𝑦 ∈ ∪ 𝐴))
26	25	expd 419	. . . . . . . 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 3202	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧 → suc 𝑦 ∈ ∪ 𝐴))
30	20, 29	syl5bi 245	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → (𝑦 ∈ ∪ 𝐴 → suc 𝑦 ∈ ∪ 𝐴))
31	30	ralrimiv 3104	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Lim 𝑥 → ∀𝑦 ∈ ∪ 𝐴 suc 𝑦 ∈ ∪ 𝐴)
32	31	adantl 485	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → ∀𝑦 ∈ ∪ 𝐴 suc 𝑦 ∈ ∪ 𝐴)
33		dflim4 7627	. 2 ⊢ (Lim ∪ 𝐴 ↔ (Ord ∪ 𝐴 ∧ ∅ ∈ ∪ 𝐴 ∧ ∀𝑦 ∈ ∪ 𝐴 suc 𝑦 ∈ ∪ 𝐴))
34	10, 19, 32, 33	syl3anbrc 1345	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → Lim ∪ 𝐴)

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)