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Theorem fin4i 10293
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 10292 . . 3 (𝐴 ∈ FinIV → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐴𝑥𝐴)))
21ibi 267 . 2 (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥𝐴𝑥𝐴))
3 relen 8944 . . . . 5 Rel ≈
43brrelex1i 5733 . . . 4 (𝑋𝐴𝑋 ∈ V)
54adantl 483 . . 3 ((𝑋𝐴𝑋𝐴) → 𝑋 ∈ V)
6 psseq1 4088 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
7 breq1 5152 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
86, 7anbi12d 632 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴𝑥𝐴) ↔ (𝑋𝐴𝑋𝐴)))
98spcegv 3588 . . 3 (𝑋 ∈ V → ((𝑋𝐴𝑋𝐴) → ∃𝑥(𝑥𝐴𝑥𝐴)))
105, 9mpcom 38 . 2 ((𝑋𝐴𝑋𝐴) → ∃𝑥(𝑥𝐴𝑥𝐴))
112, 10nsyl3 138 1 ((𝑋𝐴𝑋𝐴) → ¬ 𝐴 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  wpss 3950   class class class wbr 5149  cen 8936  FinIVcfin4 10275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-en 8940  df-fin4 10282
This theorem is referenced by:  fin4en1  10304  ssfin4  10305  ominf4  10307  isfin4p1  10310
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