Theorem dif1enen 7003

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 6875	. . 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 4063	. . . . . . 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 6910	. . . . 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 3474	. . . . . 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 621	. . 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 4063	. . . . . . . . . 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 7002	. . . . . . . 8 |- ( ( m e. om /\ A ~~ suc m /\ C e. A ) -> ( A \ { C } ) ~~ m )
23	15, 20, 21, 22	syl3anc 1250	. . . . . . 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 6898	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> B ~~ A )
27		entr 6899	. . . . . . . . . 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 7002	. . . . . . . . 9 |- ( ( m e. om /\ B ~~ suc m /\ D e. B ) -> ( B \ { D } ) ~~ m )
32	15, 28, 30, 31	syl3anc 1250	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( B \ { D } ) ~~ m )
33	32	ensymd 6898	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> m ~~ ( B \ { D } ) )
34		entr 6899	. . . . . . 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 2625	. . . 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 4670	. . . 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 800	. 2 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
42	3, 41	rexlimddv 2630	1 |- ( ph -> ( A \ { C } ) ~~ ( B \ { D } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2178   E.wrex 2487   \ cdif 3171   (/)c0 3468   {csn 3643   class class class wbr 4059   suc csuc 4430   omcom 4656   ~~ cen 6848   Fincfn 6850

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-er 6643  df-en 6851  df-fin 6853

This theorem is referenced by:  fisseneq  7057