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Theorem fin4i 9708
Description: Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
fin4i ((𝑋𝐴𝑋𝐴) → ¬ 𝐴 ∈ FinIV)

Proof of Theorem fin4i
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isfin4 9707 . . 3 (𝐴 ∈ FinIV → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐴𝑥𝐴)))
21ibi 268 . 2 (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥𝐴𝑥𝐴))
3 relen 8502 . . . . 5 Rel ≈
43brrelex1i 5601 . . . 4 (𝑋𝐴𝑋 ∈ V)
54adantl 482 . . 3 ((𝑋𝐴𝑋𝐴) → 𝑋 ∈ V)
6 psseq1 4061 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
7 breq1 5060 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
86, 7anbi12d 630 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴𝑥𝐴) ↔ (𝑋𝐴𝑋𝐴)))
98spcegv 3594 . . 3 (𝑋 ∈ V → ((𝑋𝐴𝑋𝐴) → ∃𝑥(𝑥𝐴𝑥𝐴)))
105, 9mpcom 38 . 2 ((𝑋𝐴𝑋𝐴) → ∃𝑥(𝑥𝐴𝑥𝐴))
112, 10nsyl3 140 1 ((𝑋𝐴𝑋𝐴) → ¬ 𝐴 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  wpss 3934   class class class wbr 5057  cen 8494  FinIVcfin4 9690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-en 8498  df-fin4 9697
This theorem is referenced by:  fin4en1  9719  ssfin4  9720  ominf4  9722  isfin4p1  9725
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