Theorem fmelpw1o 7556

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 859, which translate to 1o and (/) respectively by iftrue 3626 and iffalse 3629, giving pwtrufal 16763).

As proved in if0ab 3622, the associated element of ~P 1o is the extension, in ~P 1o, of the formula ph. (Contributed by BJ, 15-Aug-2024.) (Proof shortened by BJ, 5-May-2026.)

Assertion

Ref	Expression
fmelpw1o	|- if ( ph , 1o , (/) ) e. ~P 1o

Proof of Theorem fmelpw1o

Step	Hyp	Ref	Expression
1		1oex 6654	. 2 |- 1o e. _V
2		if0elpw 4270	. 2 |- ( 1o e. _V -> if ( ph , 1o , (/) ) e. ~P 1o )
3	1, 2	ax-mp 5	1 |- if ( ph , 1o , (/) ) e. ~P 1o

Syntax hints:   e. wcel 2203   _Vcvv 2812   (/)c0 3507   ifcif 3619   ~Pcpw 3668   1oc1o 6639

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-uni 3914  df-tr 4208  df-iord 4486  df-on 4488  df-suc 4491  df-1o 6646

This theorem is referenced by:  bj-charfun  16569  pw1map  16761