Theorem pw1ne0 7205

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 4150	. 2 |- (/) e. ~P 1o
2	1	ne0ii 3424	1 |- ~P 1o =/= (/)

Syntax hints:   =/= wne 2340   (/)c0 3414   ~Pcpw 3566   1oc1o 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568

This theorem is referenced by:  pw1nel3  7208  sucpw1nel3  7210