Theorem fmelpw1o 7464

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 858, which translate to 1o and (/) respectively by iftrue 3610 and iffalse 3613, giving pwtrufal 16598).

As proved in if0ab 16401, 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 6589	. . 3 |- 1o e. _V
2		0ex 4216	. . 3 |- (/) e. _V
3	1, 2	ifelpwun 4580	. 2 |- if ( ph , 1o , (/) ) e. ~P ( 1o u. (/) )
4		un0 3528	. . 3 |- ( 1o u. (/) ) = 1o
5	4	pweqi 3656	. 2 |- ~P ( 1o u. (/) ) = ~P 1o
6	3, 5	eleqtri 2306	1 |- if ( ph , 1o , (/) ) e. ~P 1o

Syntax hints:   e. wcel 2202   u. cun 3198   (/)c0 3494   ifcif 3605   ~Pcpw 3652   1oc1o 6574

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468  df-1o 6581

This theorem is referenced by:  bj-charfun  16402  pw1map  16596