Theorem ssorduni 7791

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 4871	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
2		ssel 3925	. . . . . . . . 9 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → 𝑦 ∈ On))
3		onelss 6404	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
4	2, 3	syl6 36	. . . . . . . 8 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)))
5		anc2r 564	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)) → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → (𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴))))
6	4, 5	syl 18	. . . . . . 7 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → (𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴))))
7		ssuni 4893	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ⊆ ∪ 𝐴)
8	6, 7	syl8 77	. . . . . 6 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴)))
9	8	rexlimdv 3162	. . . . 5 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴))
10	1, 9	biimtrid 245	. . . 4 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ⊆ ∪ 𝐴))
11	10	ralrimiv 3154	. . 3 ⊢ (𝐴 ⊆ On → ∀𝑥 ∈ ∪ 𝐴𝑥 ⊆ ∪ 𝐴)
12		dftr3 5217	. . 3 ⊢ (Tr ∪ 𝐴 ↔ ∀𝑥 ∈ ∪ 𝐴𝑥 ⊆ ∪ 𝐴)
13	11, 12	sylibr 237	. 2 ⊢ (𝐴 ⊆ On → Tr ∪ 𝐴)
14		onelon 6386	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On)
15	14	ex 418	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ∈ On))
16	2, 15	syl6 36	. . . . 5 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ∈ On)))
17	16	rexlimdv 3162	. . . 4 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ∈ On))
18	1, 17	biimtrid 245	. . 3 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ On))
19	18	ssrdv 3937	. 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ⊆ On)
20		ordon 7789	. . 3 ⊢ Ord On
21		trssord 6378	. . . 4 ⊢ ((Tr ∪ 𝐴 ∧ ∪ 𝐴 ⊆ On ∧ Ord On) → Ord ∪ 𝐴)
22	21	3exp 1137	. . 3 ⊢ (Tr ∪ 𝐴 → (∪ 𝐴 ⊆ On → (Ord On → Ord ∪ 𝐴)))
23	20, 22	mpii 47	. 2 ⊢ (Tr ∪ 𝐴 → (∪ 𝐴 ⊆ On → Ord ∪ 𝐴))
24	13, 19, 23	sylc 66	1 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  ∪ cuni 4867  Tr wtr 5212  Ord word 6360  Oncon0 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6364  df-on 6365

This theorem is used by:  ssonuni  7792  ssonprc  7799  orduni  7801  onsucuni  7837  limuni3  7861  onfununi  8342  tfrlem8  8385  cofon1  8674  cofon2  8675  naddcllem  8678  onssnum  10112  unialeph  10173  cfslbn  10338  hsmexlem1  10497  inaprc  10914  bdayimaon  28043  noetasuplem4  28086  noetainflem4  28090  noeta2  28140  etaslts2  28173  cutbdaybnd2lim  28176  onsupneqmaxlim0  44210  onsupnmax  44214  onsupsucismax  44265  onsucunifi  44356