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Theorem fin1ai 9703
Description: Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
fin1ai ((𝐴 ∈ FinIa𝑋𝐴) → (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin))

Proof of Theorem fin1ai
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2897 . . 3 (𝑥 = 𝑋 → (𝑥 ∈ Fin ↔ 𝑋 ∈ Fin))
2 difeq2 4090 . . . 4 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
32eleq1d 2894 . . 3 (𝑥 = 𝑋 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
41, 3orbi12d 912 . 2 (𝑥 = 𝑋 → ((𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin) ↔ (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin)))
5 isfin1a 9702 . . . 4 (𝐴 ∈ FinIa → (𝐴 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin)))
65ibi 268 . . 3 (𝐴 ∈ FinIa → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin))
76adantr 481 . 2 ((𝐴 ∈ FinIa𝑋𝐴) → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin))
8 elpw2g 5238 . . 3 (𝐴 ∈ FinIa → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
98biimpar 478 . 2 ((𝐴 ∈ FinIa𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
104, 7, 9rspcdva 3622 1 ((𝐴 ∈ FinIa𝑋𝐴) → (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135  cdif 3930  wss 3933  𝒫 cpw 4535  Fincfn 8497  FinIacfin1a 9688
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
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  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-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-pw 4537  df-fin1a 9695
This theorem is referenced by:  enfin1ai  9794  fin1a2  9825  fin1aufil  22468
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