Theorem pw1ne0 7295

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 4197	. 2 |- (/) e. ~P 1o
2	1	ne0ii 3460	1 |- ~P 1o =/= (/)

Syntax hints:   =/= wne 2367   (/)c0 3450   ~Pcpw 3605   1oc1o 6467

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4159

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607

This theorem is referenced by:  pw1nel3  7298  sucpw1nel3  7300