Theorem onintonm 4554

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 3178	. . . . . . 7 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
2		eloni 4411	. . . . . . . 8 ⊢ (𝑥 ∈ On → Ord 𝑥)
3		ordtr 4414	. . . . . . . 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 2569	. . . . 5 ⊢ (𝐴 ⊆ On → ∀𝑥 ∈ 𝐴 Tr 𝑥)
7		trint 4147	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)
8	6, 7	syl 14	. . . 4 ⊢ (𝐴 ⊆ On → Tr ∩ 𝐴)
9	8	adantr 276	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → Tr ∩ 𝐴)
10		nfv 1542	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ On
11		nfe1 1510	. . . . 5 ⊢ Ⅎ𝑥∃𝑥 𝑥 ∈ 𝐴
12	10, 11	nfan 1579	. . . 4 ⊢ Ⅎ𝑥(𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴)
13		intssuni2m 3899	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ On)
14		unon 4548	. . . . . . . 8 ⊢ ∪ On = On
15	13, 14	sseqtrdi 3232	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ On)
16	15	sseld 3183	. . . . . 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 2568	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∀𝑥 ∈ ∩ 𝐴Tr 𝑥)
20		dford3 4403	. . 3 ⊢ (Ord ∩ 𝐴 ↔ (Tr ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴Tr 𝑥))
21	9, 19, 20	sylanbrc 417	. 2 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → Ord ∩ 𝐴)
22		inteximm 4183	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
23	22	adantl 277	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ V)
24		elong 4409	. . 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 167	1 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ⊆ wss 3157  ∪ cuni 3840  ∩ cint 3875  Tr wtr 4132  Ord word 4398  Oncon0 4399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-tr 4133  df-iord 4402  df-on 4404  df-suc 4407

This theorem is referenced by:  onintrab2im  4555