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.

Step	Hyp	Ref	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
6	4, 5	wc 50	. . . . . . . 8 |- (z:(al -> *)x:al):*
7		axpow.1	. . . . . . . . 9 |- A:(al -> *)
8	7, 5	wc 50	. . . . . . . 8 |- (Ax:al):*
9	3, 6, 8	wov 72	. . . . . . 7 |- [(z:(al -> *)x:al) ==> (Ax:al)]:*
10	9	wl 66	. . . . . 6 |- \x:al [(z:(al -> *)x:al) ==> (Ax:al)]:(al -> *)
11	2, 10	wc 50	. . . . 5 |- (A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]):*
12	1, 11	simpl 22	. . . 4 |- (T., (A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)])) |= T.
13	12	ex 158	. . 3 |- T. |= [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> T.]
14	13	alrimiv 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 -> *) -> *)
17	16, 4	wc 50	. . . . . 6 |- (y:((al -> *) -> *)z:(al -> *)):*
18	3, 11, 17	wov 72	. . . . 5 |- [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]:*
19	18	wl 66	. . . 4 |- \z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]:((al -> *) -> *)
20	15, 19	wc 50	. . 3 |- (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]):*
21	1	wl 66	. . 3 |- \p:(al -> *) T.:((al -> *) -> *)
22	16, 21	weqi 76	. . . . . . . . 9 |- [y:((al -> *) -> *) = \p:(al -> *) T.]:*
23	22	id 25	. . . . . . . 8 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [y:((al -> *) -> *) = \p:(al -> *) T.]
24	16, 4, 23	ceq1 89	. . . . . . 7 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [(y:((al -> *) -> *)z:(al -> *)) = (\p:(al -> *) T.z:(al -> *))]
25		wv 64	. . . . . . . . . . 11 |- p:(al -> *):(al -> *)
26	25, 4	weqi 76	. . . . . . . . . 10 |- [p:(al -> *) = z:(al -> *)]:*
27	26, 1	eqid 83	. . . . . . . . 9 |- [p:(al -> *) = z:(al -> *)] |= [T. = T.]
28	1, 4, 27	cl 116	. . . . . . . 8 |- T. |= [(\p:(al -> *) T.z:(al -> *)) = T.]
29	22, 28	a1i 28	. . . . . . 7 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [(\p:(al -> *) T.z:(al -> *)) = T.]
30	17, 24, 29	eqtri 95	. . . . . 6 |- [y:((al -> *) -> *) = \p:(al -> *) T.] |= [(y:((al -> *) -> *)z:(al -> *)) = T.]
31	3, 11, 17, 30	oveq2 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.]]
32	18, 31	leq 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.]]
33	15, 19, 32	ceq2 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.])]
34	20, 21, 33	cla4ev 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 -> *))]))
35	14, 34	syl 16	1 |- T. |= (E.\y:((al -> *) -> *) (A.\z:(al -> *) [(A.\x:al [(z:(al -> *)x:al) ==> (Ax:al)]) ==> (y:((al -> *) -> *)z:(al -> *))]))

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)