Theorem dif1enen 6879

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 6760	. . 3 |- ( A e. Fin <-> E. n e. om A ~~ n )
3	1, 2	sylib 122	. 2 |- ( ph -> E. n e. om A ~~ n )
4		simplrr 536	. . . . . 6 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ n )
5		breq2 4007	. . . . . . 7 |- ( n = (/) -> ( A ~~ n <-> A ~~ (/) ) )
6	5	adantl 277	. . . . . 6 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> ( A ~~ n <-> A ~~ (/) ) )
7	4, 6	mpbid 147	. . . . 5 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
8		en0 6794	. . . . 5 |- ( A ~~ (/) <-> A = (/) )
9	7, 8	sylib 122	. . . 4 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
10		dif1enen.c	. . . . . 6 |- ( ph -> C e. A )
11		n0i 3428	. . . . . 6 |- ( C e. A -> -. A = (/) )
12	10, 11	syl 14	. . . . 5 |- ( ph -> -. A = (/) )
13	12	ad2antrr 488	. . . 4 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> -. A = (/) )
14	9, 13	pm2.21dd 620	. . 3 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
15		simplr 528	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> m e. om )
16		simprr 531	. . . . . . . . . 10 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> A ~~ n )
17	16	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ n )
18		breq2 4007	. . . . . . . . . 10 |- ( n = suc m -> ( A ~~ n <-> A ~~ suc m ) )
19	18	adantl 277	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( A ~~ n <-> A ~~ suc m ) )
20	17, 19	mpbid 147	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ suc m )
21	10	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> C e. A )
22		dif1en 6878	. . . . . . . 8 |- ( ( m e. om /\ A ~~ suc m /\ C e. A ) -> ( A \ { C } ) ~~ m )
23	15, 20, 21, 22	syl3anc 1238	. . . . . . 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 492	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ B )
26	25	ensymd 6782	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> B ~~ A )
27		entr 6783	. . . . . . . . . 10 |- ( ( B ~~ A /\ A ~~ suc m ) -> B ~~ suc m )
28	26, 20, 27	syl2anc 411	. . . . . . . . 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 492	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> D e. B )
31		dif1en 6878	. . . . . . . . 9 |- ( ( m e. om /\ B ~~ suc m /\ D e. B ) -> ( B \ { D } ) ~~ m )
32	15, 28, 30, 31	syl3anc 1238	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( B \ { D } ) ~~ m )
33	32	ensymd 6782	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> m ~~ ( B \ { D } ) )
34		entr 6783	. . . . . . 7 |- ( ( ( A \ { C } ) ~~ m /\ m ~~ ( B \ { D } ) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
35	23, 33, 34	syl2anc 411	. . . . . 6 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
36	35	ex 115	. . . . 5 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) -> ( n = suc m -> ( A \ { C } ) ~~ ( B \ { D } ) ) )
37	36	rexlimdva 2594	. . . 4 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> ( E. m e. om n = suc m -> ( A \ { C } ) ~~ ( B \ { D } ) ) )
38	37	imp 124	. . 3 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ E. m e. om n = suc m ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
39		nn0suc 4603	. . . 4 |- ( n e. om -> ( n = (/) \/ E. m e. om n = suc m ) )
40	39	ad2antrl 490	. . 3 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> ( n = (/) \/ E. m e. om n = suc m ) )
41	14, 38, 40	mpjaodan 798	. 2 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
42	3, 41	rexlimddv 2599	1 |- ( ph -> ( A \ { C } ) ~~ ( B \ { D } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   E.wrex 2456   \ cdif 3126   (/)c0 3422   {csn 3592   class class class wbr 4003   suc csuc 4365   omcom 4589   ~~ cen 6737   Fincfn 6739

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-er 6534  df-en 6740  df-fin 6742

This theorem is referenced by:  fisseneq  6930