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Theorem axpow 221
Description: Axiom of Power Sets. An axiom of Zermelo-Fraenkel set theory. (Contributed by Mario Carneiro, 10-Oct-2014.)
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 43 . . . . 5 |- T.:*
2 wal 134 . . . . . 6 |- A.:((al -> *) -> *)
3 wim 137 . . . . . . . 8 |- ==> :(* -> (* -> *))
4 wv 64 . . . . . . . . 9 |- z:(al -> *):(al -> *)
5 wv 64 . . . . . . . . 9 |- x:al:al
64, 5wc 50 . . . . . . . 8 |- (z:(al -> *)x:al):*
7 axpow.1 . . . . . . . . 9 |- A:(al -> *)
87, 5wc 50 . . . . . . . 8 |- (Ax:al):*
93, 6, 8wov 72 . . . . . . 7 |- [(z:(al -> *)x:al) ==> (Ax:al)]:*
109wl 66 . . . . . 6 |- \x:al [(z:(al -> *)x:al) ==> (Ax:al)]:(al -> *)
112, 10wc 50 . . . . 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 158 . . 3 |- T. |= [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> T.]
1413alrimiv 151 . 2 |- T. |= (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> T.])
15 wal 134 . . . 4 |- A.:(((al -> *) -> *) -> *)
16 wv 64 . . . . . . 7 |- y:((al -> *) -> *):((al -> *) -> *)
1716, 4wc 50 . . . . . 6 |- (y:((al -> *) -> *)z:(al -> *)):*
183, 11, 17wov 72 . . . . 5 |- [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]:*
1918wl 66 . . . 4 |- \z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]:((al -> *) -> *)
2015, 19wc 50 . . 3 |- (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]):*
211wl 66 . . 3 |- \p:(al -> *) T.:((al -> *) -> *)
2216, 21weqi 76 . . . . . . . . 9 |- [y:((al -> *) -> *) = \p:(al -> *) T.]:*
2322id 25 . . . . . . . 8 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [y:((al -> *) -> *) = \p:(al -> *) T.]
2416, 4, 23ceq1 89 . . . . . . 7 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [(y:((al -> *) -> *)z:(al -> *)) = (\p:(al -> *) T.z:(al -> *))]
25 wv 64 . . . . . . . . . . 11 |- p:(al -> *):(al -> *)
2625, 4weqi 76 . . . . . . . . . 10 |- [p:(al -> *) = z:(al -> *)]:*
2726, 1eqid 83 . . . . . . . . 9 |- [p:(al -> *) = z:(al -> *)] |= [T. = T.]
281, 4, 27cl 116 . . . . . . . 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 95 . . . . . 6 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [(y:((al -> *) -> *)z:(al -> *)) = T.]
313, 11, 17, 30oveq2 101 . . . . 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 91 . . . 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 90 . . 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 169 . 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 121  A.tal 122  E.tex 123
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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113
This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129  df-ex 131
This theorem is referenced by: (None)
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