Theorem fmelpw1o 7606

Description: With a formula 𝜑 one can associate an element of 𝒫 1_o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than ⊤ and ⊥, by nndc 863, which translate to 1_o and ∅ respectively by iftrue 3645 and iffalse 3648, giving pwtrufal 17027).

As proved in if0ab 3641, the associated element of 𝒫 1_o is the extension, in 𝒫 1_o, of the formula 𝜑. (Contributed by BJ, 15-Aug-2024.) (Proof shortened by BJ, 5-May-2026.)

Assertion

Ref	Expression
fmelpw1o	⊢ if(𝜑, 1o, ∅) ∈ 𝒫 1o

Proof of Theorem fmelpw1o

Step	Hyp	Ref	Expression
1		1oex 6695	. 2 ⊢ 1o ∈ V
2		if0elpw 4295	. 2 ⊢ (1o ∈ V → if(𝜑, 1o, ∅) ∈ 𝒫 1o)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 1o

This proof depends on syntax axioms:   ∈ wcel 2209  Vcvv 2821  ∅c0 3520  ifcif 3638  𝒫 cpw 3688  1_oc1o 6680

This proof depends on 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-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-uni 3936  df-tr 4230  df-iord 4511  df-on 4513  df-suc 4516  df-1o 6687

This theorem is used by:  bj-charfun  16833  pw1map  17025