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Theorem snnen2o 7704
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 7286 . . . 4  |-  1o  e.  om
2 php5 7703 . . . 4  |-  ( 1o  e.  om  ->  -.  1o  ~~  suc  1o )
31, 2ax-mp 5 . . 3  |-  -.  1o  ~~ 
suc  1o
4 ensn1g 7578 . . 3  |-  ( A  e.  _V  ->  { A }  ~~  1o )
5 df-2o 7129 . . . . . 6  |-  2o  =  suc  1o
65eqcomi 2454 . . . . 5  |-  suc  1o  =  2o
76breq2i 4441 . . . 4  |-  ( 1o 
~~  suc  1o  <->  1o  ~~  2o )
8 ensymb 7561 . . . . . 6  |-  ( { A }  ~~  1o  <->  1o 
~~  { A }
)
9 entr 7565 . . . . . . 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 7137 . . . 4  |-  2o  =/=  (/)
16 ensymb 7561 . . . . 5  |-  ( (/)  ~~  2o  <->  2o  ~~  (/) )
17 en0 7576 . . . . 5  |-  ( 2o 
~~  (/)  <->  2o  =  (/) )
1816, 17bitri 249 . . . 4  |-  ( (/)  ~~  2o  <->  2o  =  (/) )
1915, 18nemtbir 2769 . . 3  |-  -.  (/)  ~~  2o
20 snprc 4074 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2120biimpi 194 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
2221breq1d 4443 . . 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 1381    e. wcel 1802   _Vcvv 3093   (/)c0 3767   {csn 4010   class class class wbr 4433   suc csuc 4866   omcom 6681   1oc1o 7121   2oc2o 7122    ~~ cen 7511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-1o 7128  df-2o 7129  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517
This theorem is referenced by:  pmtrsn  16413
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