Theorem iinon 7720

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 5625	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
2	1	adantr 474	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
3		eqid 2778	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	rnmptss 6656	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On)
5		dm0rn0 5587	. . . . . 6 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅)
6		dmmptg 5886	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
7	6	eqeq1d 2780	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ 𝐴 = ∅))
8	5, 7	syl5bbr 277	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ 𝐴 = ∅))
9	8	necon3bid 3013	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ↔ 𝐴 ≠ ∅))
10	9	biimpar 471	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
11		oninton 7278	. . 3 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On)
12	4, 10, 11	syl2an2r 675	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On)
13	2, 12	eqeltrd 2859	1 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ On)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090   ⊆ wss 3792  ∅c0 4141  ∩ cint 4710  ∩ ciin 4754   ↦ cmpt 4965  dom cdm 5355  ran crn 5356  Oncon0 5976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143

This theorem is referenced by: (None)