Theorem hashge3el3dif 12508

Description: A set with size at least 3 has at least 3 different elements. In contrast to hashge2el2dif 12505, 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 4608	. . . . . . . . 9 |- (/) =/= { (/) }
2		0ex 4569	. . . . . . . . . . . 12 |- (/) e. _V
3	2	sneqr 4183	. . . . . . . . . . 11 |- ( { (/) } = { { (/) } } -> (/) = { (/) } )
4	3	necon3i 2694	. . . . . . . . . 10 |- ( (/) =/= { (/) } -> { (/) } =/= { { (/) } } )
5	1, 4	ax-mp 5	. . . . . . . . 9 |- { (/) } =/= { { (/) } }
6		snex 4678	. . . . . . . . . 10 |- { (/) } e. _V
7		snnzg 4133	. . . . . . . . . 10 |- ( { (/) } e. _V -> { { (/) } } =/= (/) )
8	6, 7	ax-mp 5	. . . . . . . . 9 |- { { (/) } } =/= (/)
9	1, 5, 8	3pm3.2i 1172	. . . . . . . 8 |- ( (/) =/= { (/) } /\ { (/) } =/= { { (/) } } /\ { { (/) } } =/= (/) )
10		snex 4678	. . . . . . . . . 10 |- { { (/) } } e. _V
11	2, 6, 10	3pm3.2i 1172	. . . . . . . . 9 |- ( (/) e. _V /\ { (/) } e. _V /\ { { (/) } } e. _V )
12		hashtpg 12507	. . . . . . . . 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 2467	. . . . . 6 |- 3 = ( # ` { (/) , { (/) } , { { (/) } } } )
16	15	a1i 11	. . . . 5 |- ( D e. V -> 3 = ( # ` { (/) , { (/) } , { { (/) } } } ) )
17	16	breq1d 4449	. . . 4 |- ( D e. V -> ( 3 <_ ( # ` D ) <-> ( # ` { (/) , { (/) } , { { (/) } } } ) <_ ( # ` D ) ) )
18		tpfi 7788	. . . . 5 |- { (/) , { (/) } , { { (/) } } } e. Fin
19		hashdom 12430	. . . . 5 |- ( ( { (/) , { (/) } , { { (/) } } } e. Fin /\ D e. V ) -> ( ( # ` { (/) , { (/) } , { { (/) } } } ) <_ ( # ` D ) <-> { (/) , { (/) } , { { (/) } } } ~<_ D ) )
20	18, 19	mpan 668	. . . 4 |- ( D e. V -> ( ( # ` { (/) , { (/) } , { { (/) } } } ) <_ ( # ` D ) <-> { (/) , { (/) } , { { (/) } } } ~<_ D ) )
21	17, 20	bitrd 253	. . 3 |- ( D e. V -> ( 3 <_ ( # ` D ) <-> { (/) , { (/) } , { { (/) } } } ~<_ D ) )
22		brdomg 7519	. . . 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 2725	. . . . . . . . . . 11 |- ( { (/) } e. _V -> (/) =/= { { (/) } } )
25	6, 24	ax-mp 5	. . . . . . . . . 10 |- (/) =/= { { (/) } }
26	1, 25, 5	3pm3.2i 1172	. . . . . . . . 9 |- ( (/) =/= { (/) } /\ (/) =/= { { (/) } } /\ { (/) } =/= { { (/) } } )
27	26	a1i 11	. . . . . . . 8 |- ( ( D e. V /\ f : { (/) , { (/) } , { { (/) } } } -1-1-> D ) -> ( (/) =/= { (/) } /\ (/) =/= { { (/) } } /\ { (/) } =/= { { (/) } } ) )
28		simpr 459	. . . . . . . 8 |- ( ( D e. V /\ f : { (/) , { (/) } , { { (/) } } } -1-1-> D ) -> f : { (/) , { (/) } , { { (/) } } } -1-1-> D )
29	23, 27, 28	f1dom3el3dif 6151	. . . . . . 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 433	. . . . . 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 1727	. . . . 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 427	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 367   /\ w3a 971   = wceq 1398   E.wex 1617   e. wcel 1823   =/= wne 2649   E.wrex 2805   _Vcvv 3106   (/)c0 3783   {csn 4016   {ctp 4020   class class class wbr 4439   -1-1->wf1 5567   ` cfv 5570   ~<_ cdom 7507   Fincfn 7509   <_ cle 9618   3c3 10582   #chash 12387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388

This theorem is referenced by:  pmtr3ncom  16699