Theorem prfidceq 7125

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 6994	. . . . 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 3684	. . . . . 6 |- { A } = { A , A }
6		preq2 3750	. . . . . 6 |- ( A = B -> { A , A } = { A , B } )
7	5, 6	eqtrid 2275	. . . . 5 |- ( A = B -> { A } = { A , B } )
8	7	eleq1d 2299	. . . 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 2409	. . 3 |- ( -. A = B -> A =/= B )
13		prfidisj 7124	. . 3 |- ( ( A e. C /\ B e. C /\ A =/= B ) -> { A , B } e. Fin )
14	1, 11, 12, 13	syl2an3an 1334	. 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 2237	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
17	16	dcbid 845	. . . . . 6 |- ( x = A -> (DECID x = y <-> DECID A = y ) )
18		eqeq2 2240	. . . . . . 7 |- ( y = B -> ( A = y <-> A = B ) )
19	18	dcbid 845	. . . . . 6 |- ( y = B -> (DECID A = y <-> DECID A = B ) )
20	17, 19	rspc2v 2922	. . . . 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 843	. . 3 |- (DECID A = B -> ( A = B \/ -. A = B ) )
24	22, 23	syl 14	. 2 |- ( ph -> ( A = B \/ -. A = B ) )
25	10, 14, 24	mpjaodan 805	1 |- ( ph -> { A , B } e. Fin )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2201   =/= wne 2401   A.wral 2509   {csn 3670   {cpr 3671   Fincfn 6914

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-1o 6587  df-er 6707  df-en 6915  df-fin 6917

This theorem is referenced by:  tpfidceq  7127  perfectlem2  15753