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Theorem fin1ai 10226
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 2825 . . 3 (𝑥 = 𝑋 → (𝑥 ∈ Fin ↔ 𝑋 ∈ Fin))
2 difeq2 4075 . . . 4 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
32eleq1d 2822 . . 3 (𝑥 = 𝑋 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
41, 3orbi12d 917 . 2 (𝑥 = 𝑋 → ((𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin) ↔ (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin)))
5 isfin1a 10225 . . . 4 (𝐴 ∈ FinIa → (𝐴 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin)))
65ibi 266 . . 3 (𝐴 ∈ FinIa → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin))
76adantr 481 . 2 ((𝐴 ∈ FinIa𝑋𝐴) → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin))
8 elpw2g 5300 . . 3 (𝐴 ∈ FinIa → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
98biimpar 478 . 2 ((𝐴 ∈ FinIa𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
104, 7, 9rspcdva 3581 1 ((𝐴 ∈ FinIa𝑋𝐴) → (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3063  cdif 3906  wss 3909  𝒫 cpw 4559  Fincfn 8880  FinIacfin1a 10211
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5255
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3064  df-rab 3407  df-v 3446  df-dif 3912  df-in 3916  df-ss 3926  df-pw 4561  df-fin1a 10218
This theorem is referenced by:  enfin1ai  10317  fin1a2  10348  fin1aufil  23279
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