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Theorem snnen2o 7597
Description: A singleton  { A } is never equinumerous with the ordinal number 2. This holds for proper singletons ( A  e.  _V) as well as for singletons being the empty set ( A  e/  _V). (Contributed by AV, 6-Aug-2019.)
Assertion
Ref Expression
snnen2o  |-  -.  { A }  ~~  2o

Proof of Theorem snnen2o
StepHypRef Expression
1 1onn 7175 . . . 4  |-  1o  e.  om
2 php5 7596 . . . 4  |-  ( 1o  e.  om  ->  -.  1o  ~~  suc  1o )
31, 2ax-mp 5 . . 3  |-  -.  1o  ~~ 
suc  1o
4 ensn1g 7471 . . 3  |-  ( A  e.  _V  ->  { A }  ~~  1o )
5 df-2o 7018 . . . . . 6  |-  2o  =  suc  1o
65eqcomi 2463 . . . . 5  |-  suc  1o  =  2o
76breq2i 4395 . . . 4  |-  ( 1o 
~~  suc  1o  <->  1o  ~~  2o )
8 ensymb 7454 . . . . . 6  |-  ( { A }  ~~  1o  <->  1o 
~~  { A }
)
9 entr 7458 . . . . . . 7  |-  ( ( 1o  ~~  { A }  /\  { A }  ~~  2o )  ->  1o  ~~  2o )
109ex 434 . . . . . 6  |-  ( 1o 
~~  { A }  ->  ( { A }  ~~  2o  ->  1o  ~~  2o ) )
118, 10sylbi 195 . . . . 5  |-  ( { A }  ~~  1o  ->  ( { A }  ~~  2o  ->  1o  ~~  2o ) )
1211con3rr3 136 . . . 4  |-  ( -.  1o  ~~  2o  ->  ( { A }  ~~  1o  ->  -.  { A }  ~~  2o ) )
137, 12sylnbi 306 . . 3  |-  ( -.  1o  ~~  suc  1o  ->  ( { A }  ~~  1o  ->  -.  { A }  ~~  2o ) )
143, 4, 13mpsyl 63 . 2  |-  ( A  e.  _V  ->  -.  { A }  ~~  2o )
15 2on0 7026 . . . 4  |-  2o  =/=  (/)
16 ensymb 7454 . . . . 5  |-  ( (/)  ~~  2o  <->  2o  ~~  (/) )
17 en0 7469 . . . . 5  |-  ( 2o 
~~  (/)  <->  2o  =  (/) )
1816, 17bitri 249 . . . 4  |-  ( (/)  ~~  2o  <->  2o  =  (/) )
1915, 18nemtbir 2774 . . 3  |-  -.  (/)  ~~  2o
20 snprc 4034 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2120biimpi 194 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
2221breq1d 4397 . . 3  |-  ( -.  A  e.  _V  ->  ( { A }  ~~  2o 
<->  (/)  ~~  2o ) )
2319, 22mtbiri 303 . 2  |-  ( -.  A  e.  _V  ->  -. 
{ A }  ~~  2o )
2414, 23pm2.61i 164 1  |-  -.  { A }  ~~  2o
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3065   (/)c0 3732   {csn 3972   class class class wbr 4387   suc csuc 4816   omcom 6573   1oc1o 7010   2oc2o 7011    ~~ cen 7404
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-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  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-br 4388  df-opab 4446  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-om 6574  df-1o 7017  df-2o 7018  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410
This theorem is referenced by:  pmtrsn  16124
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