Theorem fmelpw1o 7455

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 3608 and iffalse 3611, giving pwtrufal 16534).

As proved in if0ab 16337, 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 6585	. . 3 |- 1o e. _V
2		0ex 4214	. . 3 |- (/) e. _V
3	1, 2	ifelpwun 4578	. 2 |- if ( ph , 1o , (/) ) e. ~P ( 1o u. (/) )
4		un0 3526	. . 3 |- ( 1o u. (/) ) = 1o
5	4	pweqi 3654	. 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 3196   (/)c0 3492   ifcif 3603   ~Pcpw 3650   1oc1o 6570

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 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

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 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892  df-tr 4186  df-iord 4461  df-on 4463  df-suc 4466  df-1o 6577

This theorem is referenced by:  bj-charfun  16338  pw1map  16532