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Theorem pw1ne1 7227
Description: The power set of  1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
Assertion
Ref Expression
pw1ne1  |-  ~P 1o  =/=  1o

Proof of Theorem pw1ne1
StepHypRef Expression
1 pw1on 7224 . . . 4  |-  ~P 1o  e.  On
21onirri 4542 . . 3  |-  -.  ~P 1o  e.  ~P 1o
3 df1o2 6429 . . . . 5  |-  1o  =  { (/) }
4 pwpw0ss 3804 . . . . . . . 8  |-  { (/) ,  { (/) } }  C_  ~P { (/) }
53pweqi 3579 . . . . . . . 8  |-  ~P 1o  =  ~P { (/) }
64, 5sseqtrri 3190 . . . . . . 7  |-  { (/) ,  { (/) } }  C_  ~P 1o
7 0ex 4130 . . . . . . . 8  |-  (/)  e.  _V
8 p0ex 4188 . . . . . . . 8  |-  { (/) }  e.  _V
97, 8prss 3748 . . . . . . 7  |-  ( (
(/)  e.  ~P 1o  /\ 
{ (/) }  e.  ~P 1o )  <->  { (/) ,  { (/) } }  C_  ~P 1o )
106, 9mpbir 146 . . . . . 6  |-  ( (/)  e.  ~P 1o  /\  { (/)
}  e.  ~P 1o )
1110simpri 113 . . . . 5  |-  { (/) }  e.  ~P 1o
123, 11eqeltri 2250 . . . 4  |-  1o  e.  ~P 1o
13 eleq1 2240 . . . 4  |-  ( ~P 1o  =  1o  ->  ( ~P 1o  e.  ~P 1o 
<->  1o  e.  ~P 1o ) )
1412, 13mpbiri 168 . . 3  |-  ( ~P 1o  =  1o  ->  ~P 1o  e.  ~P 1o )
152, 14mto 662 . 2  |-  -.  ~P 1o  =  1o
1615neir 2350 1  |-  ~P 1o  =/=  1o
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353    e. wcel 2148    =/= wne 2347    C_ wss 3129   (/)c0 3422   ~Pcpw 3575   {csn 3592   {cpr 3593   1oc1o 6409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371  df-1o 6416
This theorem is referenced by:  pw1nel3  7229  sucpw1nel3  7231
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