Theorem iinon 7324

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 5300	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
2	1	adantr 480	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
3		eqid 2610	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	rnmptss 6299	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On)
5	4	adantr 480	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On)
6		dm0rn0 5263	. . . . . 6 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅)
7		dmmptg 5549	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
8	7	eqeq1d 2612	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ 𝐴 = ∅))
9	6, 8	syl5bbr 273	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ 𝐴 = ∅))
10	9	necon3bid 2826	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ↔ 𝐴 ≠ ∅))
11	10	biimpar 501	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
12		oninton 6892	. . 3 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On)
13	5, 11, 12	syl2anc 691	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On)
14	2, 13	eqeltrd 2688	1 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ On)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  ∩ ciin 4456   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  Oncon0 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812

This theorem is referenced by: (None)