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Theorem iinon 8142
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 5863 . . 3 (∀𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
21adantr 480 . 2 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
3 eqid 2738 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43rnmptss 6978 . . 3 (∀𝑥𝐴 𝐵 ∈ On → ran (𝑥𝐴𝐵) ⊆ On)
5 dm0rn0 5823 . . . . . 6 (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅)
6 dmmptg 6134 . . . . . . 7 (∀𝑥𝐴 𝐵 ∈ On → dom (𝑥𝐴𝐵) = 𝐴)
76eqeq1d 2740 . . . . . 6 (∀𝑥𝐴 𝐵 ∈ On → (dom (𝑥𝐴𝐵) = ∅ ↔ 𝐴 = ∅))
85, 7bitr3id 284 . . . . 5 (∀𝑥𝐴 𝐵 ∈ On → (ran (𝑥𝐴𝐵) = ∅ ↔ 𝐴 = ∅))
98necon3bid 2987 . . . 4 (∀𝑥𝐴 𝐵 ∈ On → (ran (𝑥𝐴𝐵) ≠ ∅ ↔ 𝐴 ≠ ∅))
109biimpar 477 . . 3 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥𝐴𝐵) ≠ ∅)
11 oninton 7622 . . 3 ((ran (𝑥𝐴𝐵) ⊆ On ∧ ran (𝑥𝐴𝐵) ≠ ∅) → ran (𝑥𝐴𝐵) ∈ On)
124, 10, 11syl2an2r 681 . 2 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥𝐴𝐵) ∈ On)
132, 12eqeltrd 2839 1 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2108  wne 2942  wral 3063  wss 3883  c0 4253   cint 4876   ciin 4922  cmpt 5153  dom cdm 5580  ran crn 5581  Oncon0 6251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-fun 6420  df-fn 6421  df-f 6422
This theorem is referenced by: (None)
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