Theorem prfidceq 7098

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 6975	. . . . 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 3680	. . . . . 6 |- { A } = { A , A }
6		preq2 3744	. . . . . 6 |- ( A = B -> { A , A } = { A , B } )
7	5, 6	eqtrid 2274	. . . . 5 |- ( A = B -> { A } = { A , B } )
8	7	eleq1d 2298	. . . 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 2408	. . 3 |- ( -. A = B -> A =/= B )
13		prfidisj 7097	. . 3 |- ( ( A e. C /\ B e. C /\ A =/= B ) -> { A , B } e. Fin )
14	1, 11, 12, 13	syl2an3an 1332	. 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 2236	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
17	16	dcbid 843	. . . . . 6 |- ( x = A -> (DECID x = y <-> DECID A = y ) )
18		eqeq2 2239	. . . . . . 7 |- ( y = B -> ( A = y <-> A = B ) )
19	18	dcbid 843	. . . . . 6 |- ( y = B -> (DECID A = y <-> DECID A = B ) )
20	17, 19	rspc2v 2920	. . . . 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 841	. . 3 |- (DECID A = B -> ( A = B \/ -. A = B ) )
24	22, 23	syl 14	. 2 |- ( ph -> ( A = B \/ -. A = B ) )
25	10, 14, 24	mpjaodan 803	1 |- ( ph -> { A , B } e. Fin )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   {csn 3666   {cpr 3667   Fincfn 6895

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1o 6568  df-er 6688  df-en 6896  df-fin 6898

This theorem is referenced by:  tpfidceq  7100  perfectlem2  15682