Theorem fmelpw1o 15742

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 853, which translate to 1o and (/) respectively by iftrue 3576 and iffalse 3579, giving pwtrufal 15934).

As proved in if0ab 15741, 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 6510	. . 3 |- 1o e. _V
2		0ex 4171	. . 3 |- (/) e. _V
3	1, 2	ifelpwun 4530	. 2 |- if ( ph , 1o , (/) ) e. ~P ( 1o u. (/) )
4		un0 3494	. . 3 |- ( 1o u. (/) ) = 1o
5	4	pweqi 3620	. 2 |- ~P ( 1o u. (/) ) = ~P 1o
6	3, 5	eleqtri 2280	1 |- if ( ph , 1o , (/) ) e. ~P 1o

Syntax hints:   e. wcel 2176   u. cun 3164   (/)c0 3460   ifcif 3571   ~Pcpw 3616   1oc1o 6495

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415  df-suc 4418  df-1o 6502

This theorem is referenced by:  bj-charfun  15743