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Theorem fidcen 16379
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.
StepHypRef Expression
1 isfi 6920 . . . 4 |- ( A e. Fin <-> E. x e. om A ~~ x )
21biimpi 120 . . 3 |- ( A e. Fin -> E. x e. om A ~~ x )
32adantr 276 . 2 |- ( ( A e. Fin /\ B e. Fin ) -> E. x e. om A ~~ x )
4 isfi 6920 . . . . 5 |- ( B e. Fin <-> E. y e. om B ~~ y )
54biimpi 120 . . . 4 |- ( B e. Fin -> E. y e. om B ~~ y )
65ad2antlr 489 . . 3 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) -> E. y e. om B ~~ y )
7 simplrl 535 . . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> x e. om )
8 simprl 529 . . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> y e. om )
9 nndceq 6653 . . . . . . 7 |- ( ( x e. om /\ y e. om ) -> DECID x = y )
107, 8, 9syl2anc 411 . . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> DECID x = y )
11 exmiddc 841 . . . . . 6 |- (DECID x = y -> ( x = y \/ -. x = y ) )
1210, 11syl 14 . . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> ( x = y \/ -. x = y ) )
13 simplrr 536 . . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> A ~~ x )
14 simplrr 536 . . . . . . . . . 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 )
1614, 15breqtrrd 4111 . . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ x = y ) -> B ~~ x )
1716ensymd 6943 . . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ x = y ) -> x ~~ B )
18 entr 6944 . . . . . . . 8 |- ( ( A ~~ x /\ x ~~ B ) -> A ~~ B )
1913, 17, 18syl2an2r 597 . . . . . . 7 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ x = y ) -> A ~~ B )
2019ex 115 . . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> ( x = y -> A ~~ B ) )
2113ensymd 6943 . . . . . . . . . . 11 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> x ~~ A )
22 entr 6944 . . . . . . . . . . 11 |- ( ( x ~~ A /\ A ~~ B ) -> x ~~ B )
2321, 22sylan 283 . . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> x ~~ B )
24 simplrr 536 . . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> B ~~ y )
25 entr 6944 . . . . . . . . . 10 |- ( ( x ~~ B /\ B ~~ y ) -> x ~~ y )
2623, 24, 25syl2anc 411 . . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> x ~~ y )
27 simplrl 535 . . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> y e. om )
28 nneneq 7026 . . . . . . . . . 10 |- ( ( x e. om /\ y e. om ) -> ( x ~~ y <-> x = y ) )
297, 27, 28syl2an2r 597 . . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> ( x ~~ y <-> x = y ) )
3026, 29mpbid 147 . . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) /\ A ~~ B ) -> x = y )
3130ex 115 . . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> ( A ~~ B -> x = y ) )
3231con3d 634 . . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> ( -. x = y -> -. A ~~ B ) )
3320, 32orim12d 791 . . . . 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 ) ) )
3412, 33mpd 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 840 . . . 4 |- (DECID A ~~ B <-> ( A ~~ B \/ -. A ~~ B ) )
3634, 35sylibr 134 . . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) /\ ( y e. om /\ B ~~ y ) ) -> DECID A ~~ B )
376, 36rexlimddv 2653 . 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( x e. om /\ A ~~ x ) ) -> DECID A ~~ B )
383, 37rexlimddv 2653 1 |- ( ( A e. Fin /\ B e. Fin ) -> DECID A ~~ B )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   e. wcel 2200   E.wrex 2509   class class class wbr 4083   omcom 4682   ~~ cen 6893   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-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-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  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-er 6688  df-en 6896  df-fin 6898
This theorem is referenced by: (None)
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