Theorem euhash1 12260

Description: The size of a set is 1 in terms of existential uniqueness. (Contributed by Alexander van der Vekens, 8-Feb-2018.)

Assertion

Ref	Expression
euhash1	|- ( V e. W -> ( ( # ` V ) = 1 <-> E! a a e. V ) )

Distinct variable group:   V, a

Allowed substitution hint:   W( a)

Proof of Theorem euhash1

Step	Hyp	Ref	Expression
1		hash1 12250	. . . . . . 7 |- ( # ` 1o ) = 1
2	1	eqcomi 2462	. . . . . 6 |- 1 = ( # ` 1o )
3	2	eqeq2i 2467	. . . . 5 |- ( ( # ` V ) = 1 <-> ( # ` V ) = ( # ` 1o ) )
4		1onn 7164	. . . . . . 7 |- 1o e. om
5		nnfi 7590	. . . . . . 7 |- ( 1o e. om -> 1o e. Fin )
6	4, 5	ax-mp 5	. . . . . 6 |- 1o e. Fin
7		hashen 12205	. . . . . 6 |- ( ( V e. Fin /\ 1o e. Fin ) -> ( ( # ` V ) = ( # ` 1o ) <-> V ~~ 1o ) )
8	6, 7	mpan2 671	. . . . 5 |- ( V e. Fin -> ( ( # ` V ) = ( # ` 1o ) <-> V ~~ 1o ) )
9	3, 8	syl5bb 257	. . . 4 |- ( V e. Fin -> ( ( # ` V ) = 1 <-> V ~~ 1o ) )
10	9	a1d 25	. . 3 |- ( V e. Fin -> ( V e. W -> ( ( # ` V ) = 1 <-> V ~~ 1o ) ) )
11		hashinf 12195	. . . . 5 |- ( ( V e. W /\ -. V e. Fin ) -> ( # ` V ) = +oo )
12		eqeq1 2453	. . . . . . 7 |- ( ( # ` V ) = +oo -> ( ( # ` V ) = 1 <-> +oo = 1 ) )
13		1re 9472	. . . . . . . . . 10 |- 1 e. RR
14		renepnf 9518	. . . . . . . . . 10 |- ( 1 e. RR -> 1 =/= +oo )
15	13, 14	ax-mp 5	. . . . . . . . 9 |- 1 =/= +oo
16		df-ne 2643	. . . . . . . . . 10 |- ( 1 =/= +oo <-> -. 1 = +oo )
17		pm2.21 108	. . . . . . . . . 10 |- ( -. 1 = +oo -> ( 1 = +oo -> V ~~ 1o ) )
18	16, 17	sylbi 195	. . . . . . . . 9 |- ( 1 =/= +oo -> ( 1 = +oo -> V ~~ 1o ) )
19	15, 18	ax-mp 5	. . . . . . . 8 |- ( 1 = +oo -> V ~~ 1o )
20	19	eqcoms 2461	. . . . . . 7 |- ( +oo = 1 -> V ~~ 1o )
21	12, 20	syl6bi 228	. . . . . 6 |- ( ( # ` V ) = +oo -> ( ( # ` V ) = 1 -> V ~~ 1o ) )
22		hasheni 12206	. . . . . . . 8 |- ( V ~~ 1o -> ( # ` V ) = ( # ` 1o ) )
23	1	eqeq2i 2467	. . . . . . . . . 10 |- ( ( # ` V ) = ( # ` 1o ) <-> ( # ` V ) = 1 )
24	23	biimpi 194	. . . . . . . . 9 |- ( ( # ` V ) = ( # ` 1o ) -> ( # ` V ) = 1 )
25	24	a1d 25	. . . . . . . 8 |- ( ( # ` V ) = ( # ` 1o ) -> ( ( # ` V ) = +oo -> ( # ` V ) = 1 ) )
26	22, 25	syl 16	. . . . . . 7 |- ( V ~~ 1o -> ( ( # ` V ) = +oo -> ( # ` V ) = 1 ) )
27	26	com12 31	. . . . . 6 |- ( ( # ` V ) = +oo -> ( V ~~ 1o -> ( # ` V ) = 1 ) )
28	21, 27	impbid 191	. . . . 5 |- ( ( # ` V ) = +oo -> ( ( # ` V ) = 1 <-> V ~~ 1o ) )
29	11, 28	syl 16	. . . 4 |- ( ( V e. W /\ -. V e. Fin ) -> ( ( # ` V ) = 1 <-> V ~~ 1o ) )
30	29	expcom 435	. . 3 |- ( -. V e. Fin -> ( V e. W -> ( ( # ` V ) = 1 <-> V ~~ 1o ) ) )
31	10, 30	pm2.61i 164	. 2 |- ( V e. W -> ( ( # ` V ) = 1 <-> V ~~ 1o ) )
32		euen1b 7466	. 2 |- ( V ~~ 1o <-> E! a a e. V )
33	31, 32	syl6bb 261	1 |- ( V e. W -> ( ( # ` V ) = 1 <-> E! a a e. V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1757   E!weu 2259   =/= wne 2641   class class class wbr 4376   ` cfv 5502   omcom 6562   1oc1o 6999   ~~ cen 7393   Fincfn 7396   RRcr 9368   1c1 9370   +oocpnf 9502   #chash 12190

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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446

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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-card 8196  df-cda 8424  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-hash 12191

This theorem is referenced by:  frg2wot1  30774