Theorem dif1enen 6858

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 6739	. . 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 531	. . . . . 6 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ n )
5		breq2 3993	. . . . . . 7 |- ( n = (/) -> ( A ~~ n <-> A ~~ (/) ) )
6	5	adantl 275	. . . . . 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 6773	. . . . 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 3420	. . . . . 6 |- ( C e. A -> -. A = (/) )
12	10, 11	syl 14	. . . . 5 |- ( ph -> -. A = (/) )
13	12	ad2antrr 485	. . . 4 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> -. A = (/) )
14	9, 13	pm2.21dd 615	. . 3 |- ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
15		simplr 525	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> m e. om )
16		simprr 527	. . . . . . . . . 10 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> A ~~ n )
17	16	ad2antrr 485	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ n )
18		breq2 3993	. . . . . . . . . 10 |- ( n = suc m -> ( A ~~ n <-> A ~~ suc m ) )
19	18	adantl 275	. . . . . . . . 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 489	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> C e. A )
22		dif1en 6857	. . . . . . . 8 |- ( ( m e. om /\ A ~~ suc m /\ C e. A ) -> ( A \ { C } ) ~~ m )
23	15, 20, 21, 22	syl3anc 1233	. . . . . . 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 489	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ B )
26	25	ensymd 6761	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> B ~~ A )
27		entr 6762	. . . . . . . . . 10 |- ( ( B ~~ A /\ A ~~ suc m ) -> B ~~ suc m )
28	26, 20, 27	syl2anc 409	. . . . . . . . 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 489	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> D e. B )
31		dif1en 6857	. . . . . . . . 9 |- ( ( m e. om /\ B ~~ suc m /\ D e. B ) -> ( B \ { D } ) ~~ m )
32	15, 28, 30, 31	syl3anc 1233	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( B \ { D } ) ~~ m )
33	32	ensymd 6761	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> m ~~ ( B \ { D } ) )
34		entr 6762	. . . . . . 7 |- ( ( ( A \ { C } ) ~~ m /\ m ~~ ( B \ { D } ) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
35	23, 33, 34	syl2anc 409	. . . . . 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 2587	. . . 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 4588	. . . 4 |- ( n e. om -> ( n = (/) \/ E. m e. om n = suc m ) )
40	39	ad2antrl 487	. . 3 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> ( n = (/) \/ E. m e. om n = suc m ) )
41	14, 38, 40	mpjaodan 793	. 2 |- ( ( ph /\ ( n e. om /\ A ~~ n ) ) -> ( A \ { C } ) ~~ ( B \ { D } ) )
42	3, 41	rexlimddv 2592	1 |- ( ph -> ( A \ { C } ) ~~ ( B \ { D } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   E.wrex 2449   \ cdif 3118   (/)c0 3414   {csn 3583   class class class wbr 3989   suc csuc 4350   omcom 4574   ~~ cen 6716   Fincfn 6718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719  df-fin 6721

This theorem is referenced by:  fisseneq  6909