Theorem fmelpw1o 15536

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 852, which translate to 1_o and ∅ respectively by iftrue 3567 and iffalse 3570, giving pwtrufal 15728).

As proved in if0ab 15535, the associated element of 𝒫 1_o is the extension, in 𝒫 1_o, of the formula 𝜑. (Contributed by BJ, 15-Aug-2024.)

Assertion

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

Proof of Theorem fmelpw1o

Step	Hyp	Ref	Expression
1		1oex 6491	. . 3 ⊢ 1o ∈ V
2		0ex 4161	. . 3 ⊢ ∅ ∈ V
3	1, 2	ifelpwun 4519	. 2 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 (1o ∪ ∅)
4		un0 3485	. . 3 ⊢ (1o ∪ ∅) = 1o
5	4	pweqi 3610	. 2 ⊢ 𝒫 (1o ∪ ∅) = 𝒫 1o
6	3, 5	eleqtri 2271	1 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 1o

Syntax hints:   ∈ wcel 2167   ∪ cun 3155  ∅c0 3451  ifcif 3562  𝒫 cpw 3606  1_oc1o 6476

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-tr 4133  df-iord 4402  df-on 4404  df-suc 4407  df-1o 6483

This theorem is referenced by:  bj-charfun  15537