Theorem ssorduni 7778

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 6400	. . . . . . . . 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 3161	. . . . 5 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴))
10	1, 9	biimtrid 245	. . . 4 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ⊆ ∪ 𝐴))
11	10	ralrimiv 3153	. . 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 6382	. . . . . . 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 3161	. . . 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 7776	. . 3 ⊢ Ord On
21		trssord 6374	. . . 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 3076  ∃wrex 3086   ⊆ wss 3899  ∪ cuni 4867  Tr wtr 5212  Ord word 6356  Oncon0 6357

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 2732  ax-sep 5251  ax-pr 5398

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361

This theorem is used by:  ssonuni  7779  ssonprc  7786  orduni  7788  onsucuni  7824  limuni3  7848  onfununi  8330  tfrlem8  8373  cofon1  8660  cofon2  8661  naddcllem  8664  onssnum  10043  unialeph  10104  cfslbn  10269  hsmexlem1  10428  inaprc  10845  bdayimaon  27929  noetasuplem4  27972  noetainflem4  27976  noeta2  28026  etaslts2  28059  cutbdaybnd2lim  28062  onsupneqmaxlim0  44065  onsupnmax  44069  onsupsucismax  44120  onsucunifi  44211