Theorem prfidceq 7229

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 7097	. . . . 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 3722	. . . . . 6 |- { A } = { A , A }
6		preq2 3788	. . . . . 6 |- ( A = B -> { A , A } = { A , B } )
7	5, 6	eqtrid 2283	. . . . 5 |- ( A = B -> { A } = { A , B } )
8	7	eleq1d 2307	. . . 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 2428	. . 3 |- ( -. A = B -> A =/= B )
13		prfidisj 7228	. . 3 |- ( ( A e. C /\ B e. C /\ A =/= B ) -> { A , B } e. Fin )
14	1, 11, 12, 13	syl2an3an 1339	. 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 2245	. . . . . . 7 |- ( x = A -> ( x = y <-> A = y ) )
17	16	dcbid 850	. . . . . 6 |- ( x = A -> (DECID x = y <-> DECID A = y ) )
18		eqeq2 2248	. . . . . . 7 |- ( y = B -> ( A = y <-> A = B ) )
19	18	dcbid 850	. . . . . 6 |- ( y = B -> (DECID A = y <-> DECID A = B ) )
20	17, 19	rspc2v 2943	. . . . 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 415	. . . 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 848	. . 3 |- (DECID A = B -> ( A = B \/ -. A = B ) )
24	22, 23	syl 14	. 2 |- ( ph -> ( A = B \/ -. A = B ) )
25	10, 14, 24	mpjaodan 810	1 |- ( ph -> { A , B } e. Fin )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   {csn 3708   {cpr 3709   Fincfn 7016

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1o 6681  df-er 6801  df-en 7017  df-fin 7019

This theorem is referenced by:  tpfidceq  7231  perfectlem2  16097