Theorem fmelpw1o 13341

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 837, which translate to 1_o and ∅ respectively by iftrue 3510 and iffalse 3513, giving pwtrufal 13530).

As proved in if0ab 13340, 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 6365	. . 3 ⊢ 1o ∈ V
2		0ex 4091	. . 3 ⊢ ∅ ∈ V
3	1, 2	ifelpwun 4441	. 2 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 (1o ∪ ∅)
4		un0 3427	. . 3 ⊢ (1o ∪ ∅) = 1o
5	4	pweqi 3547	. 2 ⊢ 𝒫 (1o ∪ ∅) = 𝒫 1o
6	3, 5	eleqtri 2232	1 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 1o

Syntax hints:   ∈ wcel 2128   ∪ cun 3100  ∅c0 3394  ifcif 3505  𝒫 cpw 3543  1_oc1o 6350

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-iord 4325  df-on 4327  df-suc 4330  df-1o 6357

This theorem is referenced by:  bj-charfun  13342