Theorem hashtpgim 11299

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 3717	. . . . 5 |- { A , B , C } = ( { A , B } u. { C } )
2	1	fveq2i 5698	. . . 4 |- (♯ ` { A , B , C } ) = (♯ ` ( { A , B } u. { C } ) )
3		simpl1 1031	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> A e. U )
4		simpl2 1032	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> B e. V )
5		simpr1 1034	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> A =/= B )
6		prfidisj 7234	. . . . . 6 |- ( ( A e. U /\ B e. V /\ A =/= B ) -> { A , B } e. Fin )
7	3, 4, 5, 6	syl3anc 1278	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> { A , B } e. Fin )
8		simpl3 1033	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> C e. W )
9		snfig 7103	. . . . . 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 3723	. . . . . . 7 |- { C } = { C , C }
12	11	ineq2i 3429	. . . . . 6 |- ( { A , B } i^i { C } ) = ( { A , B } i^i { C , C } )
13		simpr3 1036	. . . . . . . 8 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> C =/= A )
14	13	necomd 2506	. . . . . . 7 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> A =/= C )
15		simpr2 1035	. . . . . . 7 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> B =/= C )
16		disjpr2 3773	. . . . . . 7 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= C /\ B =/= C ) ) -> ( { A , B } i^i { C , C } ) = (/) )
17	14, 15, 14, 15, 16	syl22anc 1279	. . . . . 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 2283	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( { A , B } i^i { C } ) = (/) )
19		hashun 11247	. . . . 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 1278	. . . 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 2283	. . 3 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> (♯ ` { A , B , C } ) = ( (♯ ` { A , B } ) + (♯ ` { C } ) ) )
22		hashprg 11251	. . . . . . 7 |- ( ( A e. U /\ B e. V ) -> ( A =/= B <-> (♯ ` { A , B } ) = 2 ) )
23	3, 4, 22	syl2anc 415	. . . . . 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 11239	. . . . . 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 6103	. . . 4 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( (♯ ` { A , B } ) + (♯ ` { C } ) ) = ( 2 + 1 ) )
28		2p1e3 9439	. . . 4 |- ( 2 + 1 ) = 3
29	27, 28	eqtrdi 2287	. . 3 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( (♯ ` { A , B } ) + (♯ ` { C } ) ) = 3 )
30	21, 29	eqtrd 2271	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3709   {cpr 3710   {ctp 3711   ` cfv 5377  (class class class)co 6085   Fincfn 7022   1c1 8180   + caddc 8182   2c2 9356   3c3 9357  ♯chash 11216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9306  df-2 9364  df-3 9365  df-n0 9566  df-z 9647  df-uz 9924  df-fz 10414  df-ihash 11217

This theorem is used by:  hashtpg  11301