Theorem hashge3el3dif 12286

Description: A set with size at least 3 has at least 3 different elements. In contrast to hashge2el2dif 12283, which has an elementary proof, the dominance relation and 1-1 functions from a set with three elements which are known to be different are used to prove this theorem. Although there is also an elementary proof for this theorem, it might be much longer. After all, this proof should be kept because it can be used as template for proofs for higher cardinalities. (Contributed by AV, 20-Mar-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
hashge3el3dif	|- ( ( D e. V /\ 3 <_ ( # ` D ) ) -> E. x e. D E. y e. D E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) )

Distinct variable group:   x, D, y, z

Allowed substitution hints:   V( x, y, z)

Proof of Theorem hashge3el3dif

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nep0 4558	. . . . . . . . 9 |- (/) =/= { (/) }
2		0ex 4517	. . . . . . . . . . . 12 |- (/) e. _V
3	2	sneqr 4135	. . . . . . . . . . 11 |- ( { (/) } = { { (/) } } -> (/) = { (/) } )
4	3	necon3i 2686	. . . . . . . . . 10 |- ( (/) =/= { (/) } -> { (/) } =/= { { (/) } } )
5	1, 4	ax-mp 5	. . . . . . . . 9 |- { (/) } =/= { { (/) } }
6		snex 4628	. . . . . . . . . 10 |- { (/) } e. _V
7		snnzg 4087	. . . . . . . . . 10 |- ( { (/) } e. _V -> { { (/) } } =/= (/) )
8	6, 7	ax-mp 5	. . . . . . . . 9 |- { { (/) } } =/= (/)
9	1, 5, 8	3pm3.2i 1166	. . . . . . . 8 |- ( (/) =/= { (/) } /\ { (/) } =/= { { (/) } } /\ { { (/) } } =/= (/) )
10		snex 4628	. . . . . . . . . 10 |- { { (/) } } e. _V
11	2, 6, 10	3pm3.2i 1166	. . . . . . . . 9 |- ( (/) e. _V /\ { (/) } e. _V /\ { { (/) } } e. _V )
12		hashtpg 12285	. . . . . . . . 9 |- ( ( (/) e. _V /\ { (/) } e. _V /\ { { (/) } } e. _V ) -> ( ( (/) =/= { (/) } /\ { (/) } =/= { { (/) } } /\ { { (/) } } =/= (/) ) <-> ( # ` { (/) , { (/) } , { { (/) } } } ) = 3 ) )
13	11, 12	ax-mp 5	. . . . . . . 8 |- ( ( (/) =/= { (/) } /\ { (/) } =/= { { (/) } } /\ { { (/) } } =/= (/) ) <-> ( # ` { (/) , { (/) } , { { (/) } } } ) = 3 )
14	9, 13	mpbi 208	. . . . . . 7 |- ( # ` { (/) , { (/) } , { { (/) } } } ) = 3
15	14	eqcomi 2463	. . . . . 6 |- 3 = ( # ` { (/) , { (/) } , { { (/) } } } )
16	15	a1i 11	. . . . 5 |- ( D e. V -> 3 = ( # ` { (/) , { (/) } , { { (/) } } } ) )
17	16	breq1d 4397	. . . 4 |- ( D e. V -> ( 3 <_ ( # ` D ) <-> ( # ` { (/) , { (/) } , { { (/) } } } ) <_ ( # ` D ) ) )
18		tpfi 7685	. . . . 5 |- { (/) , { (/) } , { { (/) } } } e. Fin
19		hashdom 12241	. . . . 5 |- ( ( { (/) , { (/) } , { { (/) } } } e. Fin /\ D e. V ) -> ( ( # ` { (/) , { (/) } , { { (/) } } } ) <_ ( # ` D ) <-> { (/) , { (/) } , { { (/) } } } ~<_ D ) )
20	18, 19	mpan 670	. . . 4 |- ( D e. V -> ( ( # ` { (/) , { (/) } , { { (/) } } } ) <_ ( # ` D ) <-> { (/) , { (/) } , { { (/) } } } ~<_ D ) )
21	17, 20	bitrd 253	. . 3 |- ( D e. V -> ( 3 <_ ( # ` D ) <-> { (/) , { (/) } , { { (/) } } } ~<_ D ) )
22		brdomg 7417	. . . 4 |- ( D e. V -> ( { (/) , { (/) } , { { (/) } } } ~<_ D <-> E. f f : { (/) , { (/) } , { { (/) } } } -1-1-> D ) )
23	11	a1i 11	. . . . . . . 8 |- ( ( D e. V /\ f : { (/) , { (/) } , { { (/) } } } -1-1-> D ) -> ( (/) e. _V /\ { (/) } e. _V /\ { { (/) } } e. _V ) )
24	7	necomd 2717	. . . . . . . . . . 11 |- ( { (/) } e. _V -> (/) =/= { { (/) } } )
25	6, 24	ax-mp 5	. . . . . . . . . 10 |- (/) =/= { { (/) } }
26	1, 25, 5	3pm3.2i 1166	. . . . . . . . 9 |- ( (/) =/= { (/) } /\ (/) =/= { { (/) } } /\ { (/) } =/= { { (/) } } )
27	26	a1i 11	. . . . . . . 8 |- ( ( D e. V /\ f : { (/) , { (/) } , { { (/) } } } -1-1-> D ) -> ( (/) =/= { (/) } /\ (/) =/= { { (/) } } /\ { (/) } =/= { { (/) } } ) )
28		simpr 461	. . . . . . . 8 |- ( ( D e. V /\ f : { (/) , { (/) } , { { (/) } } } -1-1-> D ) -> f : { (/) , { (/) } , { { (/) } } } -1-1-> D )
29	23, 27, 28	f1dom3el3dif 6077	. . . . . . 7 |- ( ( D e. V /\ f : { (/) , { (/) } , { { (/) } } } -1-1-> D ) -> E. x e. D E. y e. D E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) )
30	29	expcom 435	. . . . . 6 |- ( f : { (/) , { (/) } , { { (/) } } } -1-1-> D -> ( D e. V -> E. x e. D E. y e. D E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) ) )
31	30	exlimiv 1689	. . . . 5 |- ( E. f f : { (/) , { (/) } , { { (/) } } } -1-1-> D -> ( D e. V -> E. x e. D E. y e. D E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) ) )
32	31	com12 31	. . . 4 |- ( D e. V -> ( E. f f : { (/) , { (/) } , { { (/) } } } -1-1-> D -> E. x e. D E. y e. D E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) ) )
33	22, 32	sylbid 215	. . 3 |- ( D e. V -> ( { (/) , { (/) } , { { (/) } } } ~<_ D -> E. x e. D E. y e. D E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) ) )
34	21, 33	sylbid 215	. 2 |- ( D e. V -> ( 3 <_ ( # ` D ) -> E. x e. D E. y e. D E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) ) )
35	34	imp 429	1 |- ( ( D e. V /\ 3 <_ ( # ` D ) ) -> E. x e. D E. y e. D E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   E.wex 1587   e. wcel 1758   =/= wne 2642   E.wrex 2794   _Vcvv 3065   (/)c0 3732   {csn 3972   {ctp 3976   class class class wbr 4387   -1-1->wf1 5510   ` cfv 5513   ~<_ cdom 7405   Fincfn 7407   <_ cle 9517   3c3 10470   #chash 12201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-card 8207  df-cda 8435  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-n0 10678  df-z 10745  df-uz 10960  df-fz 11536  df-hash 12202

This theorem is referenced by:  pmtr3ncom  16080