Theorem hashtpgim 11242

Description: The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.) (Revised by Jim Kingdon, 17-Apr-2026.)

Assertion

Ref	Expression
hashtpgim	|- ( ( A e. U /\ B e. V /\ C e. W ) -> ( ( A =/= B /\ B =/= C /\ C =/= A ) -> (♯ ` { A , B , C } ) = 3 ) )

Proof of Theorem hashtpgim

Step	Hyp	Ref	Expression
1		df-tp 3702	. . . . 5 |- { A , B , C } = ( { A , B } u. { C } )
2	1	fveq2i 5678	. . . 4 |- (♯ ` { A , B , C } ) = (♯ ` ( { A , B } u. { C } ) )
3		simpl1 1027	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> A e. U )
4		simpl2 1028	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> B e. V )
5		simpr1 1030	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> A =/= B )
6		prfidisj 7200	. . . . . 6 |- ( ( A e. U /\ B e. V /\ A =/= B ) -> { A , B } e. Fin )
7	3, 4, 5, 6	syl3anc 1274	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> { A , B } e. Fin )
8		simpl3 1029	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> C e. W )
9		snfig 7069	. . . . . 6 |- ( C e. W -> { C } e. Fin )
10	8, 9	syl 14	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> { C } e. Fin )
11		dfsn2 3708	. . . . . . 7 |- { C } = { C , C }
12	11	ineq2i 3423	. . . . . 6 |- ( { A , B } i^i { C } ) = ( { A , B } i^i { C , C } )
13		simpr3 1032	. . . . . . . 8 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> C =/= A )
14	13	necomd 2500	. . . . . . 7 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> A =/= C )
15		simpr2 1031	. . . . . . 7 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> B =/= C )
16		disjpr2 3758	. . . . . . 7 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= C /\ B =/= C ) ) -> ( { A , B } i^i { C , C } ) = (/) )
17	14, 15, 14, 15, 16	syl22anc 1275	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( { A , B } i^i { C , C } ) = (/) )
18	12, 17	eqtrid 2279	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( { A , B } i^i { C } ) = (/) )
19		hashun 11194	. . . . 5 |- ( ( { A , B } e. Fin /\ { C } e. Fin /\ ( { A , B } i^i { C } ) = (/) ) -> (♯ ` ( { A , B } u. { C } ) ) = ( (♯ ` { A , B } ) + (♯ ` { C } ) ) )
20	7, 10, 18, 19	syl3anc 1274	. . . 4 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> (♯ ` ( { A , B } u. { C } ) ) = ( (♯ ` { A , B } ) + (♯ ` { C } ) ) )
21	2, 20	eqtrid 2279	. . 3 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> (♯ ` { A , B , C } ) = ( (♯ ` { A , B } ) + (♯ ` { C } ) ) )
22		hashprg 11198	. . . . . . 7 |- ( ( A e. U /\ B e. V ) -> ( A =/= B <-> (♯ ` { A , B } ) = 2 ) )
23	3, 4, 22	syl2anc 411	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( A =/= B <-> (♯ ` { A , B } ) = 2 ) )
24	5, 23	mpbid 147	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> (♯ ` { A , B } ) = 2 )
25		hashsng 11186	. . . . . 6 |- ( C e. W -> (♯ ` { C } ) = 1 )
26	8, 25	syl 14	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> (♯ ` { C } ) = 1 )
27	24, 26	oveq12d 6076	. . . 4 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( (♯ ` { A , B } ) + (♯ ` { C } ) ) = ( 2 + 1 ) )
28		2p1e3 9388	. . . 4 |- ( 2 + 1 ) = 3
29	27, 28	eqtrdi 2283	. . 3 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( (♯ ` { A , B } ) + (♯ ` { C } ) ) = 3 )
30	21, 29	eqtrd 2267	. 2 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> (♯ ` { A , B , C } ) = 3 )
31	30	ex 115	1 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( ( A =/= B /\ B =/= C /\ C =/= A ) -> (♯ ` { A , B , C } ) = 3 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   u. cun 3212   i^i cin 3213   (/)c0 3512   {csn 3694   {cpr 3695   {ctp 3696   ` cfv 5357  (class class class)co 6058   Fincfn 6988   1c1 8144   + caddc 8146   2c2 9305   3c3 9306  ♯chash 11163

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-ihash 11164

This theorem is referenced by:  hashtpg  11244