Theorem iinon 8341

Description: The nonempty indexed intersection of a class of ordinal numbers 𝐵(𝑥) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Assertion

Ref	Expression
iinon	⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ On)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iinon

Step	Hyp	Ref	Expression
1		dfiin3g 5951	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
2	1	adantr 486	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
3		eqid 2761	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	rnmptss 7121	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On)
5		dm0rn0 5906	. . . . . 6 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅)
6		dmmptg 6242	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
7	6	eqeq1d 2763	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ 𝐴 = ∅))
8	5, 7	bitr3id 288	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ 𝐴 = ∅))
9	8	necon3bid 3000	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ↔ 𝐴 ≠ ∅))
10	9	biimpar 483	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
11		oninton 7807	. . 3 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On)
12	4, 10, 11	syl2an2r 698	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On)
13	2, 12	eqeltrd 2861	1 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ On)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   ⊆ wss 3899  ∅c0 4279  ∩ cint 4907  ∩ ciin 4952   ↦ cmpt 5186  dom cdm 5651  ran crn 5652  Oncon0 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6364  df-on 6365  df-fun 6539  df-fn 6540  df-f 6541

This theorem is used by: (None)