Theorem prfidceq 7187

Description: A pair is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)

Hypotheses

Ref	Expression
prfidceq.a	|- ( ph -> A e. C )
prfidceq.b	|- ( ph -> B e. C )
prfidceq.dc	|- ( ph -> A. x e. C A. y e. C DECID x = y )

Assertion

Ref	Expression
prfidceq	|- ( ph -> { A , B } e. Fin )

Distinct variable groups:   x, A, y   y, B   x, C, y

Allowed substitution hints:   ph( x, y)   B( x)

Proof of Theorem prfidceq

Step	Hyp	Ref	Expression
1		prfidceq.a	. . . . 5 |- ( ph -> A e. C )
2		snfig 7055	. . . . 5 |- ( A e. C -> { A } e. Fin )
3	1, 2	syl 14	. . . 4 |- ( ph -> { A } e. Fin )
4	3	adantr 276	. . 3 |- ( ( ph /\ A = B ) -> { A } e. Fin )
5		dfsn2 3702	. . . . . 6 |- { A } = { A , A }
6		preq2 3768	. . . . . 6 |- ( A = B -> { A , A } = { A , B } )
7	5, 6	eqtrid 2277	. . . . 5 |- ( A = B -> { A } = { A , B } )
8	7	eleq1d 2301	. . . 4 |- ( A = B -> ( { A } e. Fin <-> { A , B } e. Fin ) )
9	8	adantl 277	. . 3 |- ( ( ph /\ A = B ) -> ( { A } e. Fin <-> { A , B } e. Fin ) )
10	4, 9	mpbid 147	. 2 |- ( ( ph /\ A = B ) -> { A , B } e. Fin )
11		prfidceq.b	. . 3 |- ( ph -> B e. C )
12		neqne 2420	. . 3 |- ( -. A = B -> A =/= B )
13		prfidisj 7186	. . 3 |- ( ( A e. C /\ B e. C /\ A =/= B ) -> { A , B } e. Fin )
14	1, 11, 12, 13	syl2an3an 1335	. 2 |- ( ( ph /\ -. A = B ) -> { A , B } e. Fin )
15		prfidceq.dc	. . . 4 |- ( ph -> A. x e. C A. y e. C DECID x = y )
16		eqeq1 2239	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
17	16	dcbid 846	. . . . . 6 |- ( x = A -> (DECID x = y <-> DECID A = y ) )
18		eqeq2 2242	. . . . . . 7 |- ( y = B -> ( A = y <-> A = B ) )
19	18	dcbid 846	. . . . . 6 |- ( y = B -> (DECID A = y <-> DECID A = B ) )
20	17, 19	rspc2v 2933	. . . . 5 |- ( ( A e. C /\ B e. C ) -> ( A. x e. C A. y e. C DECID x = y -> DECID A = B ) )
21	1, 11, 20	syl2anc 411	. . . 4 |- ( ph -> ( A. x e. C A. y e. C DECID x = y -> DECID A = B ) )
22	15, 21	mpd 13	. . 3 |- ( ph -> DECID A = B )
23		exmiddc 844	. . 3 |- (DECID A = B -> ( A = B \/ -. A = B ) )
24	22, 23	syl 14	. 2 |- ( ph -> ( A = B \/ -. A = B ) )
25	10, 14, 24	mpjaodan 806	1 |- ( ph -> { A , B } e. Fin )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520   {csn 3688   {cpr 3689   Fincfn 6974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-1o 6646  df-er 6766  df-en 6975  df-fin 6977

This theorem is referenced by:  tpfidceq  7189  perfectlem2  15855