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Theorem iinon 7781
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 5678 . . 3 (∀𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
21adantr 473 . 2 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
3 eqid 2778 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43rnmptss 6709 . . 3 (∀𝑥𝐴 𝐵 ∈ On → ran (𝑥𝐴𝐵) ⊆ On)
5 dm0rn0 5640 . . . . . 6 (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅)
6 dmmptg 5935 . . . . . . 7 (∀𝑥𝐴 𝐵 ∈ On → dom (𝑥𝐴𝐵) = 𝐴)
76eqeq1d 2780 . . . . . 6 (∀𝑥𝐴 𝐵 ∈ On → (dom (𝑥𝐴𝐵) = ∅ ↔ 𝐴 = ∅))
85, 7syl5bbr 277 . . . . 5 (∀𝑥𝐴 𝐵 ∈ On → (ran (𝑥𝐴𝐵) = ∅ ↔ 𝐴 = ∅))
98necon3bid 3011 . . . 4 (∀𝑥𝐴 𝐵 ∈ On → (ran (𝑥𝐴𝐵) ≠ ∅ ↔ 𝐴 ≠ ∅))
109biimpar 470 . . 3 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥𝐴𝐵) ≠ ∅)
11 oninton 7331 . . 3 ((ran (𝑥𝐴𝐵) ⊆ On ∧ ran (𝑥𝐴𝐵) ≠ ∅) → ran (𝑥𝐴𝐵) ∈ On)
124, 10, 11syl2an2r 672 . 2 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ran (𝑥𝐴𝐵) ∈ On)
132, 12eqeltrd 2866 1 ((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  wne 2967  wral 3088  wss 3829  c0 4178   cint 4749   ciin 4793  cmpt 5008  dom cdm 5407  ran crn 5408  Oncon0 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-ord 6032  df-on 6033  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196
This theorem is referenced by: (None)
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