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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
StepHypRef Expression
1 hash1 12250 . . . . . . 7  |-  ( # `  1o )  =  1
21eqcomi 2462 . . . . . 6  |-  1  =  ( # `  1o )
32eqeq2i 2467 . . . . 5  |-  ( (
# `  V )  =  1  <->  ( # `  V
)  =  ( # `  1o ) )
4 1onn 7164 . . . . . . 7  |-  1o  e.  om
5 nnfi 7590 . . . . . . 7  |-  ( 1o  e.  om  ->  1o  e.  Fin )
64, 5ax-mp 5 . . . . . 6  |-  1o  e.  Fin
7 hashen 12205 . . . . . 6  |-  ( ( V  e.  Fin  /\  1o  e.  Fin )  -> 
( ( # `  V
)  =  ( # `  1o )  <->  V  ~~  1o ) )
86, 7mpan2 671 . . . . 5  |-  ( V  e.  Fin  ->  (
( # `  V )  =  ( # `  1o ) 
<->  V  ~~  1o ) )
93, 8syl5bb 257 . . . 4  |-  ( V  e.  Fin  ->  (
( # `  V )  =  1  <->  V  ~~  1o ) )
109a1d 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 )
1513, 14ax-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 ) )
1816, 17sylbi 195 . . . . . . . . 9  |-  ( 1  =/= +oo  ->  ( 1  = +oo  ->  V  ~~  1o ) )
1915, 18ax-mp 5 . . . . . . . 8  |-  ( 1  = +oo  ->  V  ~~  1o )
2019eqcoms 2461 . . . . . . 7  |-  ( +oo  =  1  ->  V  ~~  1o )
2112, 20syl6bi 228 . . . . . 6  |-  ( (
# `  V )  = +oo  ->  ( ( # `
 V )  =  1  ->  V  ~~  1o ) )
22 hasheni 12206 . . . . . . . 8  |-  ( V 
~~  1o  ->  ( # `  V )  =  (
# `  1o )
)
231eqeq2i 2467 . . . . . . . . . 10  |-  ( (
# `  V )  =  ( # `  1o ) 
<->  ( # `  V
)  =  1 )
2423biimpi 194 . . . . . . . . 9  |-  ( (
# `  V )  =  ( # `  1o )  ->  ( # `  V
)  =  1 )
2524a1d 25 . . . . . . . 8  |-  ( (
# `  V )  =  ( # `  1o )  ->  ( ( # `  V )  = +oo  ->  ( # `  V
)  =  1 ) )
2622, 25syl 16 . . . . . . 7  |-  ( V 
~~  1o  ->  ( (
# `  V )  = +oo  ->  ( # `  V
)  =  1 ) )
2726com12 31 . . . . . 6  |-  ( (
# `  V )  = +oo  ->  ( V  ~~  1o  ->  ( # `  V
)  =  1 ) )
2821, 27impbid 191 . . . . 5  |-  ( (
# `  V )  = +oo  ->  ( ( # `
 V )  =  1  <->  V  ~~  1o ) )
2911, 28syl 16 . . . 4  |-  ( ( V  e.  W  /\  -.  V  e.  Fin )  ->  ( ( # `  V )  =  1  <-> 
V  ~~  1o )
)
3029expcom 435 . . 3  |-  ( -.  V  e.  Fin  ->  ( V  e.  W  -> 
( ( # `  V
)  =  1  <->  V  ~~  1o ) ) )
3110, 30pm2.61i 164 . 2  |-  ( V  e.  W  ->  (
( # `  V )  =  1  <->  V  ~~  1o ) )
32 euen1b 7466 . 2  |-  ( V 
~~  1o  <->  E! a  a  e.  V )
3331, 32syl6bb 261 1  |-  ( V  e.  W  ->  (
( # `  V )  =  1  <->  E! a 
a  e.  V ) )
Colors of variables: wff setvar class
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
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