Theorem pw1ne0 7581

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 4299	. 2 ⊢ ∅ ∈ 𝒫 1o
2	1	ne0ii 3531	1 ⊢ 𝒫 1o ≠ ∅

Syntax hints:   ≠ wne 2420  ∅c0 3520  𝒫 cpw 3688  1_oc1o 6674

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4257

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690

This theorem is referenced by:  pw1nel3  7584  sucpw1nel3  7586