Theorem onintonm 4664

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 3242	. . . . . . 7 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
2		eloni 4520	. . . . . . . 8 ⊢ (𝑥 ∈ On → Ord 𝑥)
3		ordtr 4523	. . . . . . . 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 2622	. . . . 5 ⊢ (𝐴 ⊆ On → ∀𝑥 ∈ 𝐴 Tr 𝑥)
7		trint 4244	. . . . 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 1581	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ On
11		nfe1 1549	. . . . 5 ⊢ Ⅎ𝑥∃𝑥 𝑥 ∈ 𝐴
12	10, 11	nfan 1618	. . . 4 ⊢ Ⅎ𝑥(𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴)
13		intssuni2m 3994	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ On)
14		unon 4658	. . . . . . . 8 ⊢ ∪ On = On
15	13, 14	sseqtrdi 3296	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ On)
16	15	sseld 3247	. . . . . 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 2621	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∀𝑥 ∈ ∩ 𝐴Tr 𝑥)
20		dford3 4512	. . 3 ⊢ (Ord ∩ 𝐴 ↔ (Tr ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴Tr 𝑥))
21	9, 19, 20	sylanbrc 421	. 2 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → Ord ∩ 𝐴)
22		inteximm 4285	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
23	22	adantl 277	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ V)
24		elong 4518	. . 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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ⊆ wss 3220  ∪ cuni 3935  ∩ cint 3970  Tr wtr 4229  Ord word 4507  Oncon0 4508

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-uni 3936  df-int 3971  df-tr 4230  df-iord 4511  df-on 4513  df-suc 4516

This theorem is used by:  onintrab2im  4665