Theorem hashtpgim 11110

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 3677	. . . . 5 |- { A , B , C } = ( { A , B } u. { C } )
2	1	fveq2i 5642	. . . 4 |- (♯ ` { A , B , C } ) = (♯ ` ( { A , B } u. { C } ) )
3		simpl1 1026	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> A e. U )
4		simpl2 1027	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> B e. V )
5		simpr1 1029	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> A =/= B )
6		prfidisj 7119	. . . . . 6 |- ( ( A e. U /\ B e. V /\ A =/= B ) -> { A , B } e. Fin )
7	3, 4, 5, 6	syl3anc 1273	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> { A , B } e. Fin )
8		simpl3 1028	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> C e. W )
9		snfig 6989	. . . . . 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 3683	. . . . . . 7 |- { C } = { C , C }
12	11	ineq2i 3405	. . . . . 6 |- ( { A , B } i^i { C } ) = ( { A , B } i^i { C , C } )
13		simpr3 1031	. . . . . . . 8 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> C =/= A )
14	13	necomd 2488	. . . . . . 7 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> A =/= C )
15		simpr2 1030	. . . . . . 7 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> B =/= C )
16		disjpr2 3733	. . . . . . 7 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= C /\ B =/= C ) ) -> ( { A , B } i^i { C , C } ) = (/) )
17	14, 15, 14, 15, 16	syl22anc 1274	. . . . . 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 11069	. . . . 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 1273	. . . 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 11073	. . . . . . 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 11061	. . . . . 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 6036	. . . 4 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( (♯ ` { A , B } ) + (♯ ` { C } ) ) = ( 2 + 1 ) )
28		2p1e3 9277	. . . 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 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   u. cun 3198   i^i cin 3199   (/)c0 3494   {csn 3669   {cpr 3670   {ctp 3671   ` cfv 5326  (class class class)co 6018   Fincfn 6909   1c1 8033   + caddc 8035   2c2 9194   3c3 9195  ♯chash 11038

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-ihash 11039

This theorem is referenced by:  hashtpg  11112