Theorem fin1ai 9115

Description: Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)

Assertion

Ref	Expression
fin1ai	|- ( ( A e. FinIa /\ X C_ A ) -> ( X e. Fin \/ ( A \ X ) e. Fin ) )

Proof of Theorem fin1ai

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . 3 |- ( x = X -> ( x e. Fin <-> X e. Fin ) )
2		difeq2 3722	. . . 4 |- ( x = X -> ( A \ x ) = ( A \ X ) )
3	2	eleq1d 2686	. . 3 |- ( x = X -> ( ( A \ x ) e. Fin <-> ( A \ X ) e. Fin ) )
4	1, 3	orbi12d 746	. 2 |- ( x = X -> ( ( x e. Fin \/ ( A \ x ) e. Fin ) <-> ( X e. Fin \/ ( A \ X ) e. Fin ) ) )
5		isfin1a 9114	. . . 4 |- ( A e. FinIa -> ( A e. FinIa <-> A. x e. ~P A ( x e. Fin \/ ( A \ x ) e. Fin ) ) )
6	5	ibi 256	. . 3 |- ( A e. FinIa -> A. x e. ~P A ( x e. Fin \/ ( A \ x ) e. Fin ) )
7	6	adantr 481	. 2 |- ( ( A e. FinIa /\ X C_ A ) -> A. x e. ~P A ( x e. Fin \/ ( A \ x ) e. Fin ) )
8		elpw2g 4827	. . 3 |- ( A e. FinIa -> ( X e. ~P A <-> X C_ A ) )
9	8	biimpar 502	. 2 |- ( ( A e. FinIa /\ X C_ A ) -> X e. ~P A )
10	4, 7, 9	rspcdva 3316	1 |- ( ( A e. FinIa /\ X C_ A ) -> ( X e. Fin \/ ( A \ X ) e. Fin ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   Fincfn 7955  Fin^Iacfin1a 9100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pw 4160  df-fin1a 9107

This theorem is referenced by:  enfin1ai  9206  fin1a2  9237  fin1aufil  21736