Theorem fidcen 7158

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 7002	. . . 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 7002	. . . . 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 6734	. . . . . . 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 844	. . . . . 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 4139	. . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ x = y ) -> B ~~ x )
17	16	ensymd 7025	. . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ x = y ) -> x ~~ B )
18		entr 7026	. . . . . . . 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 7025	. . . . . . . . . . 11 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> x ~~ A )
22		entr 7026	. . . . . . . . . . 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 7026	. . . . . . . . . 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 7113	. . . . . . . . . 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 794	. . . . 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 843	. . . 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 2667	. 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) -> DECID A ~~ B )
38	3, 37	rexlimddv 2667	1 |- ( ( A e. Fin /\ B e. Fin ) -> DECID A ~~ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   e. wcel 2205   E.wrex 2523   class class class wbr 4111   omcom 4714   ~~ cen 6975   Fincfn 6977

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-er 6769  df-en 6978  df-fin 6980

This theorem is referenced by:  eqsndc  7165