Theorem fmelpw1o 14643

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 3541 and iffalse 3544, giving pwtrufal 14832).

As proved in if0ab 14642, 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 6427	. . 3 |- 1o e. _V
2		0ex 4132	. . 3 |- (/) e. _V
3	1, 2	ifelpwun 4485	. 2 |- if ( ph , 1o , (/) ) e. ~P ( 1o u. (/) )
4		un0 3458	. . 3 |- ( 1o u. (/) ) = 1o
5	4	pweqi 3581	. 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 3129   (/)c0 3424   ifcif 3536   ~Pcpw 3577   1oc1o 6412

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 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

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 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373  df-1o 6419

This theorem is referenced by:  bj-charfun  14644