Theorem fmelpw1o 13688

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 841, which translate to 1_o and ∅ respectively by iftrue 3525 and iffalse 3528, giving pwtrufal 13877).

As proved in if0ab 13687, 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 6392	. . 3 ⊢ 1o ∈ V
2		0ex 4109	. . 3 ⊢ ∅ ∈ V
3	1, 2	ifelpwun 4461	. 2 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 (1o ∪ ∅)
4		un0 3442	. . 3 ⊢ (1o ∪ ∅) = 1o
5	4	pweqi 3563	. 2 ⊢ 𝒫 (1o ∪ ∅) = 𝒫 1o
6	3, 5	eleqtri 2241	1 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 1o

Syntax hints:   ∈ wcel 2136   ∪ cun 3114  ∅c0 3409  ifcif 3520  𝒫 cpw 3559  1_oc1o 6377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349  df-1o 6384

This theorem is referenced by:  bj-charfun  13689