Theorem fidcen 7088

Description: Equinumerosity of finite sets is decidable. (Contributed by Jim Kingdon, 10-Feb-2026.)

Assertion

Ref	Expression
fidcen	|- ( ( A e. Fin /\ B e. Fin ) -> DECID A ~~ B )

Proof of Theorem fidcen

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 6934	. . . 4 |- ( A e. Fin <-> E. x e. om A ~~ x )
2	1	biimpi 120	. . 3 |- ( A e. Fin -> E. x e. om A ~~ x )
3	2	adantr 276	. 2 |- ( ( A e. Fin /\ B e. Fin ) -> E. x e. om A ~~ x )
4		isfi 6934	. . . . 5 |- ( B e. Fin <-> E. y e. om B ~~ y )
5	4	biimpi 120	. . . 4 |- ( B e. Fin -> E. y e. om B ~~ y )
6	5	ad2antlr 489	. . 3 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) -> E. y e. om B ~~ y )
7		simplrl 537	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> x e. om )
8		simprl 531	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> y e. om )
9		nndceq 6667	. . . . . . 7 |- ( ( x e. om /\ y e. om ) -> DECID x = y )
10	7, 8, 9	syl2anc 411	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> DECID x = y )
11		exmiddc 843	. . . . . 6 |- (DECID x = y -> ( x = y \/ -. x = y ) )
12	10, 11	syl 14	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> ( x = y \/ -. x = y ) )
13		simplrr 538	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> A ~~ x )
14		simplrr 538	. . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ x = y ) -> B ~~ y )
15		simpr 110	. . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ x = y ) -> x = y )
16	14, 15	breqtrrd 4116	. . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ x = y ) -> B ~~ x )
17	16	ensymd 6957	. . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ x = y ) -> x ~~ B )
18		entr 6958	. . . . . . . 8 |- ( ( A ~~ x /\ x ~~ B ) -> A ~~ B )
19	13, 17, 18	syl2an2r 599	. . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ x = y ) -> A ~~ B )
20	19	ex 115	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> ( x = y -> A ~~ B ) )
21	13	ensymd 6957	. . . . . . . . . . 11 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> x ~~ A )
22		entr 6958	. . . . . . . . . . 11 |- ( ( x ~~ A /\ A ~~ B ) -> x ~~ B )
23	21, 22	sylan 283	. . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> x ~~ B )
24		simplrr 538	. . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> B ~~ y )
25		entr 6958	. . . . . . . . . 10 |- ( ( x ~~ B /\ B ~~ y ) -> x ~~ y )
26	23, 24, 25	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> x ~~ y )
27		simplrl 537	. . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> y e. om )
28		nneneq 7043	. . . . . . . . . 10 |- ( ( x e. om /\ y e. om ) -> ( x ~~ y <-> x = y ) )
29	7, 27, 28	syl2an2r 599	. . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> ( x ~~ y <-> x = y ) )
30	26, 29	mpbid 147	. . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> x = y )
31	30	ex 115	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> ( A ~~ B -> x = y ) )
32	31	con3d 636	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> ( -. x = y -> -. A ~~ B ) )
33	20, 32	orim12d 793	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> ( ( x = y \/ -. x = y ) -> ( A ~~ B \/ -. A ~~ B ) ) )
34	12, 33	mpd 13	. . . 4 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> ( A ~~ B \/ -. A ~~ B ) )
35		df-dc 842	. . . 4 |- (DECID A ~~ B <-> ( A ~~ B \/ -. A ~~ B ) )
36	34, 35	sylibr 134	. . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> DECID A ~~ B )
37	6, 36	rexlimddv 2655	. 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) -> DECID A ~~ B )
38	3, 37	rexlimddv 2655	1 |- ( ( A e. Fin /\ B e. Fin ) -> DECID A ~~ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   e. wcel 2202   E.wrex 2511   class class class wbr 4088   omcom 4688   ~~ cen 6907   Fincfn 6909

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-er 6702  df-en 6910  df-fin 6912

This theorem is referenced by:  eqsndc  7095