Theorem fmelpw1o 13688

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 841, which translate to 1o and (/) respectively by iftrue 3525 and iffalse 3528, giving pwtrufal 13877).

As proved in if0ab 13687, 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 6392	. . 3 |- 1o e. _V
2		0ex 4109	. . 3 |- (/) e. _V
3	1, 2	ifelpwun 4461	. 2 |- if ( ph , 1o , (/) ) e. ~P ( 1o u. (/) )
4		un0 3442	. . 3 |- ( 1o u. (/) ) = 1o
5	4	pweqi 3563	. 2 |- ~P ( 1o u. (/) ) = ~P 1o
6	3, 5	eleqtri 2241	1 |- if ( ph , 1o , (/) ) e. ~P 1o

Syntax hints:   e. wcel 2136   u. cun 3114   (/)c0 3409   ifcif 3520   ~Pcpw 3559   1oc1o 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