Theorem fmelpw1o 14561

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 851, which translate to 1o and (/) respectively by iftrue 3540 and iffalse 3543, giving pwtrufal 14750).

As proved in if0ab 14560, 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 6425	. . 3 |- 1o e. _V
2		0ex 4131	. . 3 |- (/) e. _V
3	1, 2	ifelpwun 4484	. 2 |- if ( ph , 1o , (/) ) e. ~P ( 1o u. (/) )
4		un0 3457	. . 3 |- ( 1o u. (/) ) = 1o
5	4	pweqi 3580	. 2 |- ~P ( 1o u. (/) ) = ~P 1o
6	3, 5	eleqtri 2252	1 |- if ( ph , 1o , (/) ) e. ~P 1o

Syntax hints:   e. wcel 2148   u. cun 3128   (/)c0 3423   ifcif 3535   ~Pcpw 3576   1oc1o 6410

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372  df-1o 6417

This theorem is referenced by:  bj-charfun  14562