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Theorem fin1ai 9305
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 3863 . . . 4 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
32eleq1d 2822 . . 3 (𝑥 = 𝑋 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
41, 3orbi12d 748 . 2 (𝑥 = 𝑋 → ((𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin) ↔ (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin)))
5 isfin1a 9304 . . . 4 (𝐴 ∈ FinIa → (𝐴 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin)))
65ibi 256 . . 3 (𝐴 ∈ FinIa → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin))
76adantr 472 . 2 ((𝐴 ∈ FinIa𝑋𝐴) → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin))
8 elpw2g 4974 . . 3 (𝐴 ∈ FinIa → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
98biimpar 503 . 2 ((𝐴 ∈ FinIa𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
104, 7, 9rspcdva 3453 1 ((𝐴 ∈ FinIa𝑋𝐴) → (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1630  wcel 2137  wral 3048  cdif 3710  wss 3713  𝒫 cpw 4300  Fincfn 8119  FinIacfin1a 9290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rab 3057  df-v 3340  df-dif 3716  df-in 3720  df-ss 3727  df-pw 4302  df-fin1a 9297
This theorem is referenced by:  enfin1ai  9396  fin1a2  9427  fin1aufil  21935
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