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Theorem iinon 8077
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 5834 . . 3 (∀𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
21adantr 484 . 2 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
3 eqid 2737 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43rnmptss 6939 . . 3 (∀𝑥𝐴 𝐵 ∈ On → ran (𝑥𝐴𝐵) ⊆ On)
5 dm0rn0 5794 . . . . . 6 (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅)
6 dmmptg 6105 . . . . . . 7 (∀𝑥𝐴 𝐵 ∈ On → dom (𝑥𝐴𝐵) = 𝐴)
76eqeq1d 2739 . . . . . 6 (∀𝑥𝐴 𝐵 ∈ On → (dom (𝑥𝐴𝐵) = ∅ ↔ 𝐴 = ∅))
85, 7bitr3id 288 . . . . 5 (∀𝑥𝐴 𝐵 ∈ On → (ran (𝑥𝐴𝐵) = ∅ ↔ 𝐴 = ∅))
98necon3bid 2985 . . . 4 (∀𝑥𝐴 𝐵 ∈ On → (ran (𝑥𝐴𝐵) ≠ ∅ ↔ 𝐴 ≠ ∅))
109biimpar 481 . . 3 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥𝐴𝐵) ≠ ∅)
11 oninton 7579 . . 3 ((ran (𝑥𝐴𝐵) ⊆ On ∧ ran (𝑥𝐴𝐵) ≠ ∅) → ran (𝑥𝐴𝐵) ∈ On)
124, 10, 11syl2an2r 685 . 2 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥𝐴𝐵) ∈ On)
132, 12eqeltrd 2838 1 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wne 2940  wral 3061  wss 3866  c0 4237   cint 4859   ciin 4905  cmpt 5135  dom cdm 5551  ran crn 5552  Oncon0 6213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-fun 6382  df-fn 6383  df-f 6384
This theorem is referenced by: (None)
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