MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snnen2o Structured version   Visualization version   Unicode version

Theorem snnen2o 7786
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 7365 . . . 4  |-  1o  e.  om
2 php5 7785 . . . 4  |-  ( 1o  e.  om  ->  -.  1o  ~~  suc  1o )
31, 2ax-mp 5 . . 3  |-  -.  1o  ~~ 
suc  1o
4 ensn1g 7659 . . 3  |-  ( A  e.  _V  ->  { A }  ~~  1o )
5 df-2o 7208 . . . . . 6  |-  2o  =  suc  1o
65eqcomi 2470 . . . . 5  |-  suc  1o  =  2o
76breq2i 4423 . . . 4  |-  ( 1o 
~~  suc  1o  <->  1o  ~~  2o )
8 ensymb 7642 . . . . . 6  |-  ( { A }  ~~  1o  <->  1o 
~~  { A }
)
9 entr 7646 . . . . . . 7  |-  ( ( 1o  ~~  { A }  /\  { A }  ~~  2o )  ->  1o  ~~  2o )
109ex 440 . . . . . 6  |-  ( 1o 
~~  { A }  ->  ( { A }  ~~  2o  ->  1o  ~~  2o ) )
118, 10sylbi 200 . . . . 5  |-  ( { A }  ~~  1o  ->  ( { A }  ~~  2o  ->  1o  ~~  2o ) )
1211con3rr3 143 . . . 4  |-  ( -.  1o  ~~  2o  ->  ( { A }  ~~  1o  ->  -.  { A }  ~~  2o ) )
137, 12sylnbi 312 . . 3  |-  ( -.  1o  ~~  suc  1o  ->  ( { A }  ~~  1o  ->  -.  { A }  ~~  2o ) )
143, 4, 13mpsyl 65 . 2  |-  ( A  e.  _V  ->  -.  { A }  ~~  2o )
15 2on0 7216 . . . 4  |-  2o  =/=  (/)
16 ensymb 7642 . . . . 5  |-  ( (/)  ~~  2o  <->  2o  ~~  (/) )
17 en0 7657 . . . . 5  |-  ( 2o 
~~  (/)  <->  2o  =  (/) )
1816, 17bitri 257 . . . 4  |-  ( (/)  ~~  2o  <->  2o  =  (/) )
1915, 18nemtbir 2730 . . 3  |-  -.  (/)  ~~  2o
20 snprc 4047 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2120biimpi 199 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
2221breq1d 4425 . . 3  |-  ( -.  A  e.  _V  ->  ( { A }  ~~  2o 
<->  (/)  ~~  2o ) )
2319, 22mtbiri 309 . 2  |-  ( -.  A  e.  _V  ->  -. 
{ A }  ~~  2o )
2414, 23pm2.61i 169 1  |-  -.  { A }  ~~  2o
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1454    e. wcel 1897   _Vcvv 3056   (/)c0 3742   {csn 3979   class class class wbr 4415   suc csuc 5443   omcom 6718   1oc1o 7200   2oc2o 7201    ~~ cen 7591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-om 6719  df-1o 7207  df-2o 7208  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597
This theorem is referenced by:  pmtrsn  17208
  Copyright terms: Public domain W3C validator