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Theorem fin1ai 10333
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 2829 . . 3 (𝑥 = 𝑋 → (𝑥 ∈ Fin ↔ 𝑋 ∈ Fin))
2 difeq2 4120 . . . 4 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
32eleq1d 2826 . . 3 (𝑥 = 𝑋 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
41, 3orbi12d 919 . 2 (𝑥 = 𝑋 → ((𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ Fin) ↔ (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin)))
5 isfin1a 10332 . . . 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 5333 . . 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 848   = wceq 1540  wcel 2108  wral 3061  cdif 3948  wss 3951  𝒫 cpw 4600  Fincfn 8985  FinIacfin1a 10318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-pw 4602  df-fin1a 10325
This theorem is referenced by:  enfin1ai  10424  fin1a2  10455  fin1aufil  23940
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