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Theorem iinon 8324
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 5957 . . 3 (∀𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
21adantr 481 . 2 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
3 eqid 2732 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43rnmptss 7107 . . 3 (∀𝑥𝐴 𝐵 ∈ On → ran (𝑥𝐴𝐵) ⊆ On)
5 dm0rn0 5917 . . . . . 6 (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅)
6 dmmptg 6231 . . . . . . 7 (∀𝑥𝐴 𝐵 ∈ On → dom (𝑥𝐴𝐵) = 𝐴)
76eqeq1d 2734 . . . . . 6 (∀𝑥𝐴 𝐵 ∈ On → (dom (𝑥𝐴𝐵) = ∅ ↔ 𝐴 = ∅))
85, 7bitr3id 284 . . . . 5 (∀𝑥𝐴 𝐵 ∈ On → (ran (𝑥𝐴𝐵) = ∅ ↔ 𝐴 = ∅))
98necon3bid 2985 . . . 4 (∀𝑥𝐴 𝐵 ∈ On → (ran (𝑥𝐴𝐵) ≠ ∅ ↔ 𝐴 ≠ ∅))
109biimpar 478 . . 3 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥𝐴𝐵) ≠ ∅)
11 oninton 7767 . . 3 ((ran (𝑥𝐴𝐵) ⊆ On ∧ ran (𝑥𝐴𝐵) ≠ ∅) → ran (𝑥𝐴𝐵) ∈ On)
124, 10, 11syl2an2r 683 . 2 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥𝐴𝐵) ∈ On)
132, 12eqeltrd 2833 1 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  wss 3945  c0 4319   cint 4944   ciin 4992  cmpt 5225  dom cdm 5670  ran crn 5671  Oncon0 6354
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4945  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-ord 6357  df-on 6358  df-fun 6535  df-fn 6536  df-f 6537
This theorem is referenced by: (None)
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