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Theorem fin1ai 10331
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 2827 . . 3 (𝑥 = 𝑋 → (𝑥 ∈ Fin ↔ 𝑋 ∈ Fin))
2 difeq2 4130 . . . 4 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
32eleq1d 2824 . . 3 (𝑥 = 𝑋 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
41, 3orbi12d 918 . 2 (𝑥 = 𝑋 → ((𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin) ↔ (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin)))
5 isfin1a 10330 . . . 4 (𝐴 ∈ FinIa → (𝐴 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin)))
65ibi 267 . . 3 (𝐴 ∈ FinIa → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin))
76adantr 480 . 2 ((𝐴 ∈ FinIa𝑋𝐴) → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin))
8 elpw2g 5339 . . 3 (𝐴 ∈ FinIa → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
98biimpar 477 . 2 ((𝐴 ∈ FinIa𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
104, 7, 9rspcdva 3623 1 ((𝐴 ∈ FinIa𝑋𝐴) → (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  cdif 3960  wss 3963  𝒫 cpw 4605  Fincfn 8984  FinIacfin1a 10316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-pw 4607  df-fin1a 10323
This theorem is referenced by:  enfin1ai  10422  fin1a2  10453  fin1aufil  23956
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