Theorem pw1ne0 7445

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 4254	. 2 |- (/) e. ~P 1o
2	1	ne0ii 3504	1 |- ~P 1o =/= (/)

Syntax hints:   =/= wne 2402   (/)c0 3494   ~Pcpw 3652   1oc1o 6574

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-nul 4215

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654

This theorem is referenced by:  pw1nel3  7448  sucpw1nel3  7450