Theorem fmelpw1o 13841

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 846, which translate to 1_o and ∅ respectively by iftrue 3531 and iffalse 3534, giving pwtrufal 14030).

As proved in if0ab 13840, 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 6403	. . 3 ⊢ 1o ∈ V
2		0ex 4116	. . 3 ⊢ ∅ ∈ V
3	1, 2	ifelpwun 4468	. 2 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 (1o ∪ ∅)
4		un0 3448	. . 3 ⊢ (1o ∪ ∅) = 1o
5	4	pweqi 3570	. 2 ⊢ 𝒫 (1o ∪ ∅) = 𝒫 1o
6	3, 5	eleqtri 2245	1 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 1o

Syntax hints:   ∈ wcel 2141   ∪ cun 3119  ∅c0 3414  ifcif 3526  𝒫 cpw 3566  1_oc1o 6388

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356  df-1o 6395

This theorem is referenced by:  bj-charfun  13842