Theorem onintonm 4494

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 3136	. . . . . . 7 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
2		eloni 4353	. . . . . . . 8 ⊢ (𝑥 ∈ On → Ord 𝑥)
3		ordtr 4356	. . . . . . . 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 2538	. . . . 5 ⊢ (𝐴 ⊆ On → ∀𝑥 ∈ 𝐴 Tr 𝑥)
7		trint 4095	. . . . 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 1516	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ On
11		nfe1 1484	. . . . 5 ⊢ Ⅎ𝑥∃𝑥 𝑥 ∈ 𝐴
12	10, 11	nfan 1553	. . . 4 ⊢ Ⅎ𝑥(𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴)
13		intssuni2m 3848	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ On)
14		unon 4488	. . . . . . . 8 ⊢ ∪ On = On
15	13, 14	sseqtrdi 3190	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ On)
16	15	sseld 3141	. . . . . 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 2537	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∀𝑥 ∈ ∩ 𝐴Tr 𝑥)
20		dford3 4345	. . 3 ⊢ (Ord ∩ 𝐴 ↔ (Tr ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴Tr 𝑥))
21	9, 19, 20	sylanbrc 414	. 2 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → Ord ∩ 𝐴)
22		inteximm 4128	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
23	22	adantl 275	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ V)
24		elong 4351	. . 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 1480   ∈ wcel 2136  ∀wral 2444  Vcvv 2726   ⊆ wss 3116  ∪ cuni 3789  ∩ cint 3824  Tr wtr 4080  Ord word 4340  Oncon0 4341

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by:  onintrab2im  4495