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Theorem iinon 7978
 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 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ On)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iinon
StepHypRef Expression
1 dfiin3g 5805 . . 3 (∀𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
21adantr 484 . 2 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
3 eqid 2798 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43rnmptss 6873 . . 3 (∀𝑥𝐴 𝐵 ∈ On → ran (𝑥𝐴𝐵) ⊆ On)
5 dm0rn0 5765 . . . . . 6 (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅)
6 dmmptg 6068 . . . . . . 7 (∀𝑥𝐴 𝐵 ∈ On → dom (𝑥𝐴𝐵) = 𝐴)
76eqeq1d 2800 . . . . . 6 (∀𝑥𝐴 𝐵 ∈ On → (dom (𝑥𝐴𝐵) = ∅ ↔ 𝐴 = ∅))
85, 7bitr3id 288 . . . . 5 (∀𝑥𝐴 𝐵 ∈ On → (ran (𝑥𝐴𝐵) = ∅ ↔ 𝐴 = ∅))
98necon3bid 3031 . . . 4 (∀𝑥𝐴 𝐵 ∈ On → (ran (𝑥𝐴𝐵) ≠ ∅ ↔ 𝐴 ≠ ∅))
109biimpar 481 . . 3 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥𝐴𝐵) ≠ ∅)
11 oninton 7508 . . 3 ((ran (𝑥𝐴𝐵) ⊆ On ∧ ran (𝑥𝐴𝐵) ≠ ∅) → ran (𝑥𝐴𝐵) ∈ On)
124, 10, 11syl2an2r 684 . 2 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥𝐴𝐵) ∈ On)
132, 12eqeltrd 2890 1 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106   ⊆ wss 3883  ∅c0 4246  ∩ cint 4842  ∩ ciin 4886   ↦ cmpt 5114  dom cdm 5523  ran crn 5524  Oncon0 6166 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6169  df-on 6170  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340 This theorem is referenced by: (None)
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