Theorem fmelpw1o 13423

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 837, which translate to 1o and (/) respectively by iftrue 3510 and iffalse 3513, giving pwtrufal 13611).

As proved in if0ab 13422, 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 6372	. . 3 |- 1o e. _V
2		0ex 4092	. . 3 |- (/) e. _V
3	1, 2	ifelpwun 4444	. 2 |- if ( ph , 1o , (/) ) e. ~P ( 1o u. (/) )
4		un0 3427	. . 3 |- ( 1o u. (/) ) = 1o
5	4	pweqi 3547	. 2 |- ~P ( 1o u. (/) ) = ~P 1o
6	3, 5	eleqtri 2232	1 |- if ( ph , 1o , (/) ) e. ~P 1o

Syntax hints:   e. wcel 2128   u. cun 3100   (/)c0 3394   ifcif 3505   ~Pcpw 3543   1oc1o 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3774  df-tr 4064  df-iord 4327  df-on 4329  df-suc 4332  df-1o 6364

This theorem is referenced by:  bj-charfun  13424