Theorem onintonm 4433

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 3091	. . . . . . 7 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
2		eloni 4297	. . . . . . . 8 ⊢ (𝑥 ∈ On → Ord 𝑥)
3		ordtr 4300	. . . . . . . 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 2504	. . . . 5 ⊢ (𝐴 ⊆ On → ∀𝑥 ∈ 𝐴 Tr 𝑥)
7		trint 4041	. . . . 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 1508	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ On
11		nfe1 1472	. . . . 5 ⊢ Ⅎ𝑥∃𝑥 𝑥 ∈ 𝐴
12	10, 11	nfan 1544	. . . 4 ⊢ Ⅎ𝑥(𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴)
13		intssuni2m 3795	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ On)
14		unon 4427	. . . . . . . 8 ⊢ ∪ On = On
15	13, 14	sseqtrdi 3145	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ On)
16	15	sseld 3096	. . . . . 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 2503	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∀𝑥 ∈ ∩ 𝐴Tr 𝑥)
20		dford3 4289	. . 3 ⊢ (Ord ∩ 𝐴 ↔ (Tr ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴Tr 𝑥))
21	9, 19, 20	sylanbrc 413	. 2 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → Ord ∩ 𝐴)
22		inteximm 4074	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
23	22	adantl 275	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ V)
24		elong 4295	. . 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 1468   ∈ wcel 1480  ∀wral 2416  Vcvv 2686   ⊆ wss 3071  ∪ cuni 3736  ∩ cint 3771  Tr wtr 4026  Ord word 4284  Oncon0 4285

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293

This theorem is referenced by:  onintrab2im  4434