Theorem iinon 8282

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 5926	. . 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 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	rnmptss 7077	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On)
5		dm0rn0 5881	. . . . . 6 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅)
6		dmmptg 6208	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
7	6	eqeq1d 2739	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ 𝐴 = ∅))
8	5, 7	bitr3id 285	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ 𝐴 = ∅))
9	8	necon3bid 2977	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ↔ 𝐴 ≠ ∅))
10	9	biimpar 477	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
11		oninton 7750	. . 3 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On)
12	4, 10, 11	syl2an2r 686	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On)
13	2, 12	eqeltrd 2837	1 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ On)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ⊆ wss 3903  ∅c0 4287  ∩ cint 4904  ∩ ciin 4949   ↦ cmpt 5181  dom cdm 5632  ran crn 5633  Oncon0 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-fun 6502  df-fn 6503  df-f 6504

This theorem is referenced by: (None)