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Theorem fin4i 9727
 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 9726 . . 3 (𝐴 ∈ FinIV → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐴𝑥𝐴)))
21ibi 270 . 2 (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥𝐴𝑥𝐴))
3 relen 8515 . . . . 5 Rel ≈
43brrelex1i 5576 . . . 4 (𝑋𝐴𝑋 ∈ V)
54adantl 485 . . 3 ((𝑋𝐴𝑋𝐴) → 𝑋 ∈ V)
6 psseq1 4018 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
7 breq1 5037 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
86, 7anbi12d 633 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴𝑥𝐴) ↔ (𝑋𝐴𝑋𝐴)))
98spcegv 3546 . . 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 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3442   ⊊ wpss 3884   class class class wbr 5034   ≈ cen 8507  FinIVcfin4 9709 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-xp 5529  df-rel 5530  df-en 8511  df-fin4 9716 This theorem is referenced by:  fin4en1  9738  ssfin4  9739  ominf4  9741  isfin4p1  9744
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