Theorem s3sndisj 13706

Description: The singletons consisting of length 3 strings which have distinct third symbols are disjunct. (Contributed by AV, 17-May-2021.)

Assertion

Ref	Expression
s3sndisj	|- ( ( A e. X /\ B e. Y ) -> Disj c e. Z { <" A B c "> } )

Distinct variable groups:   A, c   B, c   X, c   Y, c   Z, c

Proof of Theorem s3sndisj

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 400	. . . . 5 |- ( c = d -> ( c = d \/ ( { <" A B c "> } i^i { <" A B d "> } ) = (/) ) )
2	1	a1d 25	. . . 4 |- ( c = d -> ( ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) -> ( c = d \/ ( { <" A B c "> } i^i { <" A B d "> } ) = (/) ) ) )
3		s3cli 13626	. . . . . . . . . . . 12 |- <" A B c "> e. Word _V
4		elex 3212	. . . . . . . . . . . . . . 15 |- ( A e. X -> A e. _V )
5		elex 3212	. . . . . . . . . . . . . . 15 |- ( B e. Y -> B e. _V )
6	4, 5	anim12i 590	. . . . . . . . . . . . . 14 |- ( ( A e. X /\ B e. Y ) -> ( A e. _V /\ B e. _V ) )
7		elex 3212	. . . . . . . . . . . . . . 15 |- ( d e. Z -> d e. _V )
8	7	adantl 482	. . . . . . . . . . . . . 14 |- ( ( c e. Z /\ d e. Z ) -> d e. _V )
9	6, 8	anim12i 590	. . . . . . . . . . . . 13 |- ( ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) -> ( ( A e. _V /\ B e. _V ) /\ d e. _V ) )
10		df-3an 1039	. . . . . . . . . . . . 13 |- ( ( A e. _V /\ B e. _V /\ d e. _V ) <-> ( ( A e. _V /\ B e. _V ) /\ d e. _V ) )
11	9, 10	sylibr 224	. . . . . . . . . . . 12 |- ( ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) -> ( A e. _V /\ B e. _V /\ d e. _V ) )
12		eqwrds3 13704	. . . . . . . . . . . 12 |- ( ( <" A B c "> e. Word _V /\ ( A e. _V /\ B e. _V /\ d e. _V ) ) -> ( <" A B c "> = <" A B d "> <-> ( ( # ` <" A B c "> ) = 3 /\ ( ( <" A B c "> ` 0 ) = A /\ ( <" A B c "> ` 1 ) = B /\ ( <" A B c "> ` 2 ) = d ) ) ) )
13	3, 11, 12	sylancr 695	. . . . . . . . . . 11 |- ( ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) -> ( <" A B c "> = <" A B d "> <-> ( ( # ` <" A B c "> ) = 3 /\ ( ( <" A B c "> ` 0 ) = A /\ ( <" A B c "> ` 1 ) = B /\ ( <" A B c "> ` 2 ) = d ) ) ) )
14		vex 3203	. . . . . . . . . . . . . 14 |- c e. _V
15		s3fv2 13638	. . . . . . . . . . . . . 14 |- ( c e. _V -> ( <" A B c "> ` 2 ) = c )
16	14, 15	ax-mp 5	. . . . . . . . . . . . 13 |- ( <" A B c "> ` 2 ) = c
17		simp3 1063	. . . . . . . . . . . . 13 |- ( ( ( <" A B c "> ` 0 ) = A /\ ( <" A B c "> ` 1 ) = B /\ ( <" A B c "> ` 2 ) = d ) -> ( <" A B c "> ` 2 ) = d )
18	16, 17	syl5eqr 2670	. . . . . . . . . . . 12 |- ( ( ( <" A B c "> ` 0 ) = A /\ ( <" A B c "> ` 1 ) = B /\ ( <" A B c "> ` 2 ) = d ) -> c = d )
19	18	adantl 482	. . . . . . . . . . 11 |- ( ( ( # ` <" A B c "> ) = 3 /\ ( ( <" A B c "> ` 0 ) = A /\ ( <" A B c "> ` 1 ) = B /\ ( <" A B c "> ` 2 ) = d ) ) -> c = d )
20	13, 19	syl6bi 243	. . . . . . . . . 10 |- ( ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) -> ( <" A B c "> = <" A B d "> -> c = d ) )
21	20	con3rr3 151	. . . . . . . . 9 |- ( -. c = d -> ( ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) -> -. <" A B c "> = <" A B d "> ) )
22	21	imp 445	. . . . . . . 8 |- ( ( -. c = d /\ ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) ) -> -. <" A B c "> = <" A B d "> )
23	22	neqned 2801	. . . . . . 7 |- ( ( -. c = d /\ ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) ) -> <" A B c "> =/= <" A B d "> )
24		disjsn2 4247	. . . . . . 7 |- ( <" A B c "> =/= <" A B d "> -> ( { <" A B c "> } i^i { <" A B d "> } ) = (/) )
25	23, 24	syl 17	. . . . . 6 |- ( ( -. c = d /\ ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) ) -> ( { <" A B c "> } i^i { <" A B d "> } ) = (/) )
26	25	olcd 408	. . . . 5 |- ( ( -. c = d /\ ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) ) -> ( c = d \/ ( { <" A B c "> } i^i { <" A B d "> } ) = (/) ) )
27	26	ex 450	. . . 4 |- ( -. c = d -> ( ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) -> ( c = d \/ ( { <" A B c "> } i^i { <" A B d "> } ) = (/) ) ) )
28	2, 27	pm2.61i 176	. . 3 |- ( ( ( A e. X /\ B e. Y ) /\ ( c e. Z /\ d e. Z ) ) -> ( c = d \/ ( { <" A B c "> } i^i { <" A B d "> } ) = (/) ) )
29	28	ralrimivva 2971	. 2 |- ( ( A e. X /\ B e. Y ) -> A. c e. Z A. d e. Z ( c = d \/ ( { <" A B c "> } i^i { <" A B d "> } ) = (/) ) )
30		eqidd 2623	. . . . 5 |- ( c = d -> A = A )
31		eqidd 2623	. . . . 5 |- ( c = d -> B = B )
32		id 22	. . . . 5 |- ( c = d -> c = d )
33	30, 31, 32	s3eqd 13609	. . . 4 |- ( c = d -> <" A B c "> = <" A B d "> )
34	33	sneqd 4189	. . 3 |- ( c = d -> { <" A B c "> } = { <" A B d "> } )
35	34	disjor 4634	. 2 |- (Disj c e. Z { <" A B c "> } <-> A. c e. Z A. d e. Z ( c = d \/ ( { <" A B c "> } i^i { <" A B d "> } ) = (/) ) )
36	29, 35	sylibr 224	1 |- ( ( A e. X /\ B e. Y ) -> Disj c e. Z { <" A B c "> } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   i^i cin 3573   (/)c0 3915   {csn 4177  Disj wdisj 4620   ` cfv 5888   0cc0 9936   1c1 9937   2c2 11070   3c3 11071   #chash 13117  Word cword 13291   <"cs3 13587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594

This theorem is referenced by:  fusgreghash2wspv  27199