HOLE Home Higher-Order Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HOLE Home  >  Th. List  >  axpow Unicode version

Theorem axpow 208
Description: Axiom of Power Sets. An axiom of Zermelo-Fraenkel set theory.
Hypothesis
Ref Expression
axpow.1 |- A:(al -> *)
Assertion
Ref Expression
axpow |- T. |= (E.\y:((al -> *) -> *) (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]))
Distinct variable groups:   x,y   y,A   y,z,al

Proof of Theorem axpow
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 wtru 40 . . . . 5 |- T.:*
2 wal 124 . . . . . 6 |- A.:((al -> *) -> *)
3 wim 127 . . . . . . . 8 |- ==> :(* -> (* -> *))
4 wv 58 . . . . . . . . 9 |- z:(al -> *):(al -> *)
5 wv 58 . . . . . . . . 9 |- x:al:al
64, 5wc 45 . . . . . . . 8 |- (z:(al -> *)x:al):*
7 axpow.1 . . . . . . . . 9 |- A:(al -> *)
87, 5wc 45 . . . . . . . 8 |- (Ax:al):*
93, 6, 8wov 64 . . . . . . 7 |- [(z:(al -> *)x:al) ==> (Ax:al)]:*
109wl 59 . . . . . 6 |- \x:al [(z:(al -> *)x:al) ==> (Ax:al)]:(al -> *)
112, 10wc 45 . . . . 5 |- (A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]):*
121, 11simpl 22 . . . 4 |- (T., (A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)])) |= T.
1312ex 148 . . 3 |- T. |= [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> T.]
1413alrimiv 141 . 2 |- T. |= (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> T.])
15 wal 124 . . . 4 |- A.:(((al -> *) -> *) -> *)
16 wv 58 . . . . . . 7 |- y:((al -> *) -> *):((al -> *) -> *)
1716, 4wc 45 . . . . . 6 |- (y:((al -> *) -> *)z:(al -> *)):*
183, 11, 17wov 64 . . . . 5 |- [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]:*
1918wl 59 . . . 4 |- \z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]:((al -> *) -> *)
2015, 19wc 45 . . 3 |- (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]):*
211wl 59 . . 3 |- \p:(al -> *) T.:((al -> *) -> *)
2216, 21weqi 68 . . . . . . . . 9 |- [y:((al -> *) -> *) = \p:(al -> *) T.]:*
2322id 25 . . . . . . . 8 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [y:((al -> *) -> *) = \p:(al -> *) T.]
2416, 4, 23ceq1 79 . . . . . . 7 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [(y:((al -> *) -> *)z:(al -> *)) = (\p:(al -> *) T.z:(al -> *))]
25 wv 58 . . . . . . . . . . 11 |- p:(al -> *):(al -> *)
2625, 4weqi 68 . . . . . . . . . 10 |- [p:(al -> *) = z:(al -> *)]:*
2726, 1eqid 73 . . . . . . . . 9 |- [p:(al -> *) = z:(al -> *)] |= [T. = T.]
281, 4, 27cl 106 . . . . . . . 8 |- T. |= [(\p:(al -> *) T.z:(al -> *)) = T.]
2922, 28a1i 28 . . . . . . 7 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [(\p:(al -> *) T.z:(al -> *)) = T.]
3017, 24, 29eqtri 85 . . . . . 6 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [(y:((al -> *) -> *)z:(al -> *)) = T.]
313, 11, 17, 30oveq2 91 . . . . 5 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [[(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))] = [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> T.]]
3218, 31leq 81 . . . 4 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))] = \z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> T.]]
3315, 19, 32ceq2 80 . . 3 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [(A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]) = (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> T.])]
3420, 21, 33cla4ev 159 . 2 |- (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> T.]) |= (E.\y:((al -> *) -> *) (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]))
3514, 34syl 16 1 |- T. |= (E.\y:((al -> *) -> *) (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]))
Colors of variables: type var term
Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 111  A.tal 112  E.tex 113
This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103
This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119  df-ex 121
  Copyright terms: Public domain W3C validator