Theorem hashge3el3dif 12683

Description: A set with size at least 3 has at least 3 different elements. In contrast to hashge2el2dif 12678, 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 4572	. . . . . . . . 9 |- (/) =/= { (/) }
2		0ex 4528	. . . . . . . . . . . 12 |- (/) e. _V
3	2	sneqr 4131	. . . . . . . . . . 11 |- ( { (/) } = { { (/) } } -> (/) = { (/) } )
4	3	necon3i 2675	. . . . . . . . . 10 |- ( (/) =/= { (/) } -> { (/) } =/= { { (/) } } )
5	1, 4	ax-mp 5	. . . . . . . . 9 |- { (/) } =/= { { (/) } }
6		snex 4641	. . . . . . . . . 10 |- { (/) } e. _V
7		snnzg 4080	. . . . . . . . . 10 |- ( { (/) } e. _V -> { { (/) } } =/= (/) )
8	6, 7	ax-mp 5	. . . . . . . . 9 |- { { (/) } } =/= (/)
9	1, 5, 8	3pm3.2i 1208	. . . . . . . 8 |- ( (/) =/= { (/) } /\ { (/) } =/= { { (/) } } /\ { { (/) } } =/= (/) )
10		snex 4641	. . . . . . . . . 10 |- { { (/) } } e. _V
11	2, 6, 10	3pm3.2i 1208	. . . . . . . . 9 |- ( (/) e. _V /\ { (/) } e. _V /\ { { (/) } } e. _V )
12		hashtpg 12682	. . . . . . . . 9 |- ( ( (/) e. _V /\ { (/) } e. _V /\ { { (/) } } e. _V ) -> ( ( (/) =/= { (/) } /\ { (/) } =/= { { (/) } } /\ { { (/) } } =/= (/) ) <-> ( # ` { (/) , { (/) } , { { (/) } } } ) = 3 ) )
13	11, 12	ax-mp 5	. . . . . . . 8 |- ( ( (/) =/= { (/) } /\ { (/) } =/= { { (/) } } /\ { { (/) } } =/= (/) ) <-> ( # ` { (/) , { (/) } , { { (/) } } } ) = 3 )
14	9, 13	mpbi 213	. . . . . . 7 |- ( # ` { (/) , { (/) } , { { (/) } } } ) = 3
15	14	eqcomi 2480	. . . . . 6 |- 3 = ( # ` { (/) , { (/) } , { { (/) } } } )
16	15	a1i 11	. . . . 5 |- ( D e. V -> 3 = ( # ` { (/) , { (/) } , { { (/) } } } ) )
17	16	breq1d 4405	. . . 4 |- ( D e. V -> ( 3 <_ ( # ` D ) <-> ( # ` { (/) , { (/) } , { { (/) } } } ) <_ ( # ` D ) ) )
18		tpfi 7865	. . . . 5 |- { (/) , { (/) } , { { (/) } } } e. Fin
19		hashdom 12596	. . . . 5 |- ( ( { (/) , { (/) } , { { (/) } } } e. Fin /\ D e. V ) -> ( ( # ` { (/) , { (/) } , { { (/) } } } ) <_ ( # ` D ) <-> { (/) , { (/) } , { { (/) } } } ~<_ D ) )
20	18, 19	mpan 684	. . . 4 |- ( D e. V -> ( ( # ` { (/) , { (/) } , { { (/) } } } ) <_ ( # ` D ) <-> { (/) , { (/) } , { { (/) } } } ~<_ D ) )
21	17, 20	bitrd 261	. . 3 |- ( D e. V -> ( 3 <_ ( # ` D ) <-> { (/) , { (/) } , { { (/) } } } ~<_ D ) )
22		brdomg 7597	. . . 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 2698	. . . . . . . . . . 11 |- ( { (/) } e. _V -> (/) =/= { { (/) } } )
25	6, 24	ax-mp 5	. . . . . . . . . 10 |- (/) =/= { { (/) } }
26	1, 25, 5	3pm3.2i 1208	. . . . . . . . 9 |- ( (/) =/= { (/) } /\ (/) =/= { { (/) } } /\ { (/) } =/= { { (/) } } )
27	26	a1i 11	. . . . . . . 8 |- ( ( D e. V /\ f : { (/) , { (/) } , { { (/) } } } -1-1-> D ) -> ( (/) =/= { (/) } /\ (/) =/= { { (/) } } /\ { (/) } =/= { { (/) } } ) )
28		simpr 468	. . . . . . . 8 |- ( ( D e. V /\ f : { (/) , { (/) } , { { (/) } } } -1-1-> D ) -> f : { (/) , { (/) } , { { (/) } } } -1-1-> D )
29	23, 27, 28	f1dom3el3dif 6187	. . . . . . 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 442	. . . . . 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 1784	. . . . 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 223	. . 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 223	. 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 436	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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   =/= wne 2641   E.wrex 2757   _Vcvv 3031   (/)c0 3722   {csn 3959   {ctp 3963   class class class wbr 4395   -1-1->wf1 5586   ` cfv 5589   ~<_ cdom 7585   Fincfn 7587   <_ cle 9694   3c3 10682   #chash 12553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554

This theorem is referenced by:  pmtr3ncom  17194