Theorem pw1ne0 7409

Description: The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)

Assertion

Ref	Expression
pw1ne0	|- ~P 1o =/= (/)

Proof of Theorem pw1ne0

Step	Hyp	Ref	Expression
1		0elpw 4247	. 2 |- (/) e. ~P 1o
2	1	ne0ii 3501	1 |- ~P 1o =/= (/)

Syntax hints:   =/= wne 2400   (/)c0 3491   ~Pcpw 3649   1oc1o 6553

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4209

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651

This theorem is referenced by:  pw1nel3  7412  sucpw1nel3  7414