Theorem ssorduni 7755

Description: The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. Lemma 2.7 of [Schloeder] p. 4. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
ssorduni	⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)

Proof of Theorem ssorduni

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 4875	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
2		ssel 3940	. . . . . . . . 9 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → 𝑦 ∈ On))
3		onelss 6374	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
4	2, 3	syl6 35	. . . . . . . 8 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)))
5		anc2r 554	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)) → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → (𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴))))
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → (𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴))))
7		ssuni 4896	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ⊆ ∪ 𝐴)
8	6, 7	syl8 76	. . . . . 6 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴)))
9	8	rexlimdv 3132	. . . . 5 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴))
10	1, 9	biimtrid 242	. . . 4 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ⊆ ∪ 𝐴))
11	10	ralrimiv 3124	. . 3 ⊢ (𝐴 ⊆ On → ∀𝑥 ∈ ∪ 𝐴𝑥 ⊆ ∪ 𝐴)
12		dftr3 5220	. . 3 ⊢ (Tr ∪ 𝐴 ↔ ∀𝑥 ∈ ∪ 𝐴𝑥 ⊆ ∪ 𝐴)
13	11, 12	sylibr 234	. 2 ⊢ (𝐴 ⊆ On → Tr ∪ 𝐴)
14		onelon 6357	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On)
15	14	ex 412	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ∈ On))
16	2, 15	syl6 35	. . . . 5 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ∈ On)))
17	16	rexlimdv 3132	. . . 4 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ∈ On))
18	1, 17	biimtrid 242	. . 3 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ On))
19	18	ssrdv 3952	. 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ⊆ On)
20		ordon 7753	. . 3 ⊢ Ord On
21		trssord 6349	. . . 4 ⊢ ((Tr ∪ 𝐴 ∧ ∪ 𝐴 ⊆ On ∧ Ord On) → Ord ∪ 𝐴)
22	21	3exp 1119	. . 3 ⊢ (Tr ∪ 𝐴 → (∪ 𝐴 ⊆ On → (Ord On → Ord ∪ 𝐴)))
23	20, 22	mpii 46	. 2 ⊢ (Tr ∪ 𝐴 → (∪ 𝐴 ⊆ On → Ord ∪ 𝐴))
24	13, 19, 23	sylc 65	1 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914  ∪ cuni 4871  Tr wtr 5214  Ord word 6331  Oncon0 6332

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

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

This theorem is referenced by:  ssonuni  7756  ssonprc  7763  orduni  7765  onsucuni  7803  limuni3  7828  onfununi  8310  tfrlem8  8352  cofon1  8636  cofon2  8637  naddcllem  8640  onssnum  9993  unialeph  10054  cfslbn  10220  hsmexlem1  10379  inaprc  10789  bdayimaon  27605  noetasuplem4  27648  noetainflem4  27652  noeta2  27696  etasslt2  27726  scutbdaybnd2lim  27729  onsupneqmaxlim0  43213  onsupnmax  43217  onsupsucismax  43268  onsucunifi  43359