Theorem fmelpw1o 7378

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 853, which translate to 1_o and ∅ respectively by iftrue 3580 and iffalse 3583, giving pwtrufal 16075).

As proved in if0ab 15880, 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 6523	. . 3 ⊢ 1o ∈ V
2		0ex 4179	. . 3 ⊢ ∅ ∈ V
3	1, 2	ifelpwun 4538	. 2 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 (1o ∪ ∅)
4		un0 3498	. . 3 ⊢ (1o ∪ ∅) = 1o
5	4	pweqi 3625	. 2 ⊢ 𝒫 (1o ∪ ∅) = 𝒫 1o
6	3, 5	eleqtri 2281	1 ⊢ if(𝜑, 1o, ∅) ∈ 𝒫 1o

Syntax hints:   ∈ wcel 2177   ∪ cun 3168  ∅c0 3464  ifcif 3575  𝒫 cpw 3621  1_oc1o 6508

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3857  df-tr 4151  df-iord 4421  df-on 4423  df-suc 4426  df-1o 6515

This theorem is referenced by:  bj-charfun  15881  pw1map  16073