Theorem onintonm 4501

Description: The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)

Assertion

Ref	Expression
onintonm	⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ On)

Distinct variable group:   𝑥,𝐴

Proof of Theorem onintonm

Step	Hyp	Ref	Expression
1		ssel 3141	. . . . . . 7 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
2		eloni 4360	. . . . . . . 8 ⊢ (𝑥 ∈ On → Ord 𝑥)
3		ordtr 4363	. . . . . . . 8 ⊢ (Ord 𝑥 → Tr 𝑥)
4	2, 3	syl 14	. . . . . . 7 ⊢ (𝑥 ∈ On → Tr 𝑥)
5	1, 4	syl6 33	. . . . . 6 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → Tr 𝑥))
6	5	ralrimiv 2542	. . . . 5 ⊢ (𝐴 ⊆ On → ∀𝑥 ∈ 𝐴 Tr 𝑥)
7		trint 4102	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)
8	6, 7	syl 14	. . . 4 ⊢ (𝐴 ⊆ On → Tr ∩ 𝐴)
9	8	adantr 274	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → Tr ∩ 𝐴)
10		nfv 1521	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ On
11		nfe1 1489	. . . . 5 ⊢ Ⅎ𝑥∃𝑥 𝑥 ∈ 𝐴
12	10, 11	nfan 1558	. . . 4 ⊢ Ⅎ𝑥(𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴)
13		intssuni2m 3855	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ On)
14		unon 4495	. . . . . . . 8 ⊢ ∪ On = On
15	13, 14	sseqtrdi 3195	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ On)
16	15	sseld 3146	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ On))
17	16, 2	syl6 33	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → (𝑥 ∈ ∩ 𝐴 → Ord 𝑥))
18	17, 3	syl6 33	. . . 4 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → (𝑥 ∈ ∩ 𝐴 → Tr 𝑥))
19	12, 18	ralrimi 2541	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∀𝑥 ∈ ∩ 𝐴Tr 𝑥)
20		dford3 4352	. . 3 ⊢ (Ord ∩ 𝐴 ↔ (Tr ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴Tr 𝑥))
21	9, 19, 20	sylanbrc 415	. 2 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → Ord ∩ 𝐴)
22		inteximm 4135	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
23	22	adantl 275	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ V)
24		elong 4358	. . 3 ⊢ (∩ 𝐴 ∈ V → (∩ 𝐴 ∈ On ↔ Ord ∩ 𝐴))
25	23, 24	syl 14	. 2 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → (∩ 𝐴 ∈ On ↔ Ord ∩ 𝐴))
26	21, 25	mpbird 166	1 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ On)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  Vcvv 2730   ⊆ wss 3121  ∪ cuni 3796  ∩ cint 3831  Tr wtr 4087  Ord word 4347  Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356

This theorem is referenced by:  onintrab2im  4502