Theorem pw1ne0 7489

Description: The power set of 1_o 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 4260	. 2 ⊢ ∅ ∈ 𝒫 1o
2	1	ne0ii 3506	1 ⊢ 𝒫 1o ≠ ∅

Syntax hints:   ≠ wne 2403  ∅c0 3496  𝒫 cpw 3656  1_oc1o 6618

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-nul 4220

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658

This theorem is referenced by:  pw1nel3  7492  sucpw1nel3  7494