Theorem fmelpw1o 7428

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 856, which translate to 1o and (/) respectively by iftrue 3607 and iffalse 3610, giving pwtrufal 16322).

As proved in if0ab 16127, 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 6568	. . 3 |- 1o e. _V
2		0ex 4210	. . 3 |- (/) e. _V
3	1, 2	ifelpwun 4573	. 2 |- if ( ph , 1o , (/) ) e. ~P ( 1o u. (/) )
4		un0 3525	. . 3 |- ( 1o u. (/) ) = 1o
5	4	pweqi 3653	. 2 |- ~P ( 1o u. (/) ) = ~P 1o
6	3, 5	eleqtri 2304	1 |- if ( ph , 1o , (/) ) e. ~P 1o

Syntax hints:   e. wcel 2200   u. cun 3195   (/)c0 3491   ifcif 3602   ~Pcpw 3649   1oc1o 6553

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-tr 4182  df-iord 4456  df-on 4458  df-suc 4461  df-1o 6560

This theorem is referenced by:  bj-charfun  16128  pw1map  16320