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Theorem fin4i 9566
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 9565 . . 3 (𝐴 ∈ FinIV → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐴𝑥𝐴)))
21ibi 268 . 2 (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥𝐴𝑥𝐴))
3 relen 8362 . . . . 5 Rel ≈
43brrelex1i 5494 . . . 4 (𝑋𝐴𝑋 ∈ V)
54adantl 482 . . 3 ((𝑋𝐴𝑋𝐴) → 𝑋 ∈ V)
6 psseq1 3985 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
7 breq1 4965 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
86, 7anbi12d 630 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴𝑥𝐴) ↔ (𝑋𝐴𝑋𝐴)))
98spcegv 3540 . . 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 1522  wex 1761  wcel 2081  Vcvv 3437  wpss 3860   class class class wbr 4962  cen 8354  FinIVcfin4 9548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-rel 5450  df-en 8358  df-fin4 9555
This theorem is referenced by:  fin4en1  9577  ssfin4  9578  ominf4  9580  isfin4p1  9583
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