Theorem dif1enen 7068

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 6933	. . 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 538	. . . . . 6 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ n )
5		breq2 4092	. . . . . . 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 6968	. . . . 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 3500	. . . . . 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 625	. . 3 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
15		simplr 529	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> m e. om )
16		simprr 533	. . . . . . . . . 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 4092	. . . . . . . . . 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 7067	. . . . . . . 8 |- ( ( m e. om /\ A ~~ suc m /\ C e. A ) -> ( A \ { C } ) ~~ m )
23	15, 20, 21, 22	syl3anc 1273	. . . . . . 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 6956	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> B ~~ A )
27		entr 6957	. . . . . . . . . 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 7067	. . . . . . . . 9 |- ( ( m e. om /\ B ~~ suc m /\ D e. B ) -> ( B \ { D } ) ~~ m )
32	15, 28, 30, 31	syl3anc 1273	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( B \ { D } ) ~~ m )
33	32	ensymd 6956	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> m ~~ ( B \ { D } ) )
34		entr 6957	. . . . . . 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 2650	. . . 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 4702	. . . 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 805	. 2 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
42	3, 41	rexlimddv 2655	1 |- ( ph -> ( A \ { C } ) ~~ ( B \ { D } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   E.wrex 2511   \ cdif 3197   (/)c0 3494   {csn 3669   class class class wbr 4088   suc csuc 4462   omcom 4688   ~~ cen 6906   Fincfn 6908

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-coll 4204  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-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  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 6701  df-en 6909  df-fin 6911

This theorem is referenced by:  fisseneq  7126