Theorem dif1enen 6703

Description: Subtracting one element from each of two equinumerous finite sets. (Contributed by Jim Kingdon, 5-Jun-2022.)

Hypotheses

Ref	Expression
dif1enen.a	|- ( ph -> A e. Fin )
dif1enen.ab	|- ( ph -> A ~~ B )
dif1enen.c	|- ( ph -> C e. A )
dif1enen.d	|- ( ph -> D e. B )

Assertion

Ref	Expression
dif1enen	|- ( ph -> ( A \ { C } ) ~~ ( B \ { D } ) )

Proof of Theorem dif1enen

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dif1enen.a	. . 3 |- ( ph -> A e. Fin )
2		isfi 6585	. . 3 |- ( A e. Fin <-> E. n e. om A ~~ n )
3	1, 2	sylib 121	. 2 |- ( ph -> E. n e. om A ~~ n )
4		simplrr 506	. . . . . 6 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ n )
5		breq2 3879	. . . . . . 7 |- ( n = (/) -> ( A ~~ n <-> A ~~ (/) ) )
6	5	adantl 273	. . . . . 6 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> ( A ~~ n <-> A ~~ (/) ) )
7	4, 6	mpbid 146	. . . . 5 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
8		en0 6619	. . . . 5 |- ( A ~~ (/) <-> A = (/) )
9	7, 8	sylib 121	. . . 4 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
10		dif1enen.c	. . . . . 6 |- ( ph -> C e. A )
11		n0i 3315	. . . . . 6 |- ( C e. A -> -. A = (/) )
12	10, 11	syl 14	. . . . 5 |- ( ph -> -. A = (/) )
13	12	ad2antrr 475	. . . 4 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> -. A = (/) )
14	9, 13	pm2.21dd 590	. . 3 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
15		simplr 500	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> m e. om )
16		simprr 502	. . . . . . . . . 10 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> A ~~ n )
17	16	ad2antrr 475	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ n )
18		breq2 3879	. . . . . . . . . 10 |- ( n = suc m -> ( A ~~ n <-> A ~~ suc m ) )
19	18	adantl 273	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( A ~~ n <-> A ~~ suc m ) )
20	17, 19	mpbid 146	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ suc m )
21	10	ad3antrrr 479	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> C e. A )
22		dif1en 6702	. . . . . . . 8 |- ( ( m e. om /\ A ~~ suc m /\ C e. A ) -> ( A \ { C } ) ~~ m )
23	15, 20, 21, 22	syl3anc 1184	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( A \ { C } ) ~~ m )
24		dif1enen.ab	. . . . . . . . . . . 12 |- ( ph -> A ~~ B )
25	24	ad3antrrr 479	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ B )
26	25	ensymd 6607	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> B ~~ A )
27		entr 6608	. . . . . . . . . 10 |- ( ( B ~~ A /\ A ~~ suc m ) -> B ~~ suc m )
28	26, 20, 27	syl2anc 406	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> B ~~ suc m )
29		dif1enen.d	. . . . . . . . . 10 |- ( ph -> D e. B )
30	29	ad3antrrr 479	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> D e. B )
31		dif1en 6702	. . . . . . . . 9 |- ( ( m e. om /\ B ~~ suc m /\ D e. B ) -> ( B \ { D } ) ~~ m )
32	15, 28, 30, 31	syl3anc 1184	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( B \ { D } ) ~~ m )
33	32	ensymd 6607	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> m ~~ ( B \ { D } ) )
34		entr 6608	. . . . . . 7 |- ( ( ( A \ { C } ) ~~ m /\ m ~~ ( B \ { D } ) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
35	23, 33, 34	syl2anc 406	. . . . . 6 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
36	35	ex 114	. . . . 5 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) -> ( n = suc m -> ( A \ { C } ) ~~ ( B \ { D } ) ) )
37	36	rexlimdva 2508	. . . 4 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> ( E. m e. om n = suc m -> ( A \ { C } ) ~~ ( B \ { D } ) ) )
38	37	imp 123	. . 3 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ E. m e. om n = suc m ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
39		nn0suc 4456	. . . 4 |- ( n e. om -> ( n = (/) \/ E. m e. om n = suc m ) )
40	39	ad2antrl 477	. . 3 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> ( n = (/) \/ E. m e. om n = suc m ) )
41	14, 38, 40	mpjaodan 753	. 2 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
42	3, 41	rexlimddv 2513	1 |- ( ph -> ( A \ { C } ) ~~ ( B \ { D } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 670   = wceq 1299   e. wcel 1448   E.wrex 2376   \ cdif 3018   (/)c0 3310   {csn 3474   class class class wbr 3875   suc csuc 4225   omcom 4442   ~~ cen 6562   Fincfn 6564

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-er 6359  df-en 6565  df-fin 6567

This theorem is referenced by:  fisseneq  6749