Theorem hashtpgim 11153

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 3681	. . . . 5 |- { A , B , C } = ( { A , B } u. { C } )
2	1	fveq2i 5651	. . . 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 7162	. . . . . 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 7032	. . . . . 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 3687	. . . . . . 7 |- { C } = { C , C }
12	11	ineq2i 3407	. . . . . 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 2489	. . . . . . 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 3737	. . . . . . 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 2276	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( { A , B } i^i { C } ) = (/) )
19		hashun 11112	. . . . 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 2276	. . 3 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> (♯ ` { A , B , C } ) = ( (♯ ` { A , B } ) + (♯ ` { C } ) ) )
22		hashprg 11116	. . . . . . 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 11104	. . . . . 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 6046	. . . 4 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( (♯ ` { A , B } ) + (♯ ` { C } ) ) = ( 2 + 1 ) )
28		2p1e3 9320	. . . 4 |- ( 2 + 1 ) = 3
29	27, 28	eqtrdi 2280	. . 3 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( (♯ ` { A , B } ) + (♯ ` { C } ) ) = 3 )
30	21, 29	eqtrd 2264	. 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 2202   =/= wne 2403   u. cun 3199   i^i cin 3200   (/)c0 3496   {csn 3673   {cpr 3674   {ctp 3675   ` cfv 5333  (class class class)co 6028   Fincfn 6952   1c1 8076   + caddc 8078   2c2 9237   3c3 9238  ♯chash 11081

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-2 9245  df-3 9246  df-n0 9446  df-z 9523  df-uz 9799  df-fz 10287  df-ihash 11082

This theorem is referenced by:  hashtpg  11155