Theorem fmelpw1o 7393

Description: With a formula ph one can associate an element of ~P 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than T. and F., by nndc 853, which translate to 1o and (/) respectively by iftrue 3584 and iffalse 3587, giving pwtrufal 16136).

As proved in if0ab 15941, the associated element of ~P 1o is the extension, in ~P 1o, of the formula ph. (Contributed by BJ, 15-Aug-2024.)

Assertion

Ref	Expression
fmelpw1o	|- if ( ph , 1o , (/) ) e. ~P 1o

Proof of Theorem fmelpw1o

Step	Hyp	Ref	Expression
1		1oex 6533	. . 3 |- 1o e. _V
2		0ex 4187	. . 3 |- (/) e. _V
3	1, 2	ifelpwun 4548	. 2 |- if ( ph , 1o , (/) ) e. ~P ( 1o u. (/) )
4		un0 3502	. . 3 |- ( 1o u. (/) ) = 1o
5	4	pweqi 3630	. 2 |- ~P ( 1o u. (/) ) = ~P 1o
6	3, 5	eleqtri 2282	1 |- if ( ph , 1o , (/) ) e. ~P 1o

Syntax hints:   e. wcel 2178   u. cun 3172   (/)c0 3468   ifcif 3579   ~Pcpw 3626   1oc1o 6518

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 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433  df-suc 4436  df-1o 6525

This theorem is referenced by:  bj-charfun  15942  pw1map  16134