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Theorem fin4i 9064
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 9063 . . 3 (𝐴 ∈ FinIV → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐴𝑥𝐴)))
21ibi 256 . 2 (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥𝐴𝑥𝐴))
3 relen 7904 . . . . 5 Rel ≈
43brrelexi 5118 . . . 4 (𝑋𝐴𝑋 ∈ V)
54adantl 482 . . 3 ((𝑋𝐴𝑋𝐴) → 𝑋 ∈ V)
6 psseq1 3672 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
7 breq1 4616 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
86, 7anbi12d 746 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴𝑥𝐴) ↔ (𝑋𝐴𝑋𝐴)))
98spcegv 3280 . . 3 (𝑋 ∈ V → ((𝑋𝐴𝑋𝐴) → ∃𝑥(𝑥𝐴𝑥𝐴)))
105, 9mpcom 38 . 2 ((𝑋𝐴𝑋𝐴) → ∃𝑥(𝑥𝐴𝑥𝐴))
112, 10nsyl3 133 1 ((𝑋𝐴𝑋𝐴) → ¬ 𝐴 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  wpss 3556   class class class wbr 4613  cen 7896  FinIVcfin4 9046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-en 7900  df-fin4 9053
This theorem is referenced by:  fin4en1  9075  ssfin4  9076  ominf4  9078  isfin4-3  9081
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