Theorem axun 222

Description: Axiom of Union. An axiom of Zermelo-Fraenkel set theory. (Contributed by Mario Carneiro, 10-Oct-2014.)

Hypothesis

Ref	Expression
axun.1	|- A:((al -> *) -> *)

Assertion

Ref	Expression
axun	|- T. |= (E.\y:(al -> *) (A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)]))

Distinct variable groups:   x,y   y,z   y,A   al,y,z

Proof of Theorem axun

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wtru 43	. . . . . 6 |- T.:*
2		wex 139	. . . . . . 7 |- E.:(((al -> *) -> *) -> *)
3		wan 136	. . . . . . . . 9 |- /\ :(* -> (* -> *))
4		wv 64	. . . . . . . . . 10 |- x:(al -> *):(al -> *)
5		wv 64	. . . . . . . . . 10 |- z:al:al
6	4, 5	wc 50	. . . . . . . . 9 |- (x:(al -> *)z:al):*
7		axun.1	. . . . . . . . . 10 |- A:((al -> *) -> *)
8	7, 4	wc 50	. . . . . . . . 9 |- (Ax:(al -> *)):*
9	3, 6, 8	wov 72	. . . . . . . 8 |- [(x:(al -> *)z:al) /\ (Ax:(al -> *))]:*
10	9	wl 66	. . . . . . 7 |- \x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]:((al -> *) -> *)
11	2, 10	wc 50	. . . . . 6 |- (E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]):*
12	1, 11	wct 48	. . . . 5 |- (T., (E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))])):*
13	12	trud 27	. . . 4 |- (T., (E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))])) |= T.
14	13	ex 158	. . 3 |- T. |= [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> T.]
15	14	alrimiv 151	. 2 |- T. |= (A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> T.])
16		wal 134	. . . 4 |- A.:((al -> *) -> *)
17		wim 137	. . . . . 6 |- ==> :(* -> (* -> *))
18		wv 64	. . . . . . 7 |- y:(al -> *):(al -> *)
19	18, 5	wc 50	. . . . . 6 |- (y:(al -> *)z:al):*
20	17, 11, 19	wov 72	. . . . 5 |- [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)]:*
21	20	wl 66	. . . 4 |- \z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)]:(al -> *)
22	16, 21	wc 50	. . 3 |- (A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)]):*
23	1	wl 66	. . 3 |- \p:al T.:(al -> *)
24	18, 23	weqi 76	. . . . . . . . 9 |- [y:(al -> *) = \p:al T.]:*
25	24	id 25	. . . . . . . 8 |- [y:(al -> *) = \p:al T.] |= [y:(al -> *) = \p:al T.]
26	18, 5, 25	ceq1 89	. . . . . . 7 |- [y:(al -> *) = \p:al T.] |= [(y:(al -> *)z:al) = (\p:al T.z:al)]
27	1, 5, 24	a17i 106	. . . . . . 7 |- [y:(al -> *) = \p:al T.] |= [(\p:al T.z:al) = T.]
28	19, 26, 27	eqtri 95	. . . . . 6 |- [y:(al -> *) = \p:al T.] |= [(y:(al -> *)z:al) = T.]
29	17, 11, 19, 28	oveq2 101	. . . . 5 |- [y:(al -> *) = \p:al T.] |= [[(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)] = [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> T.]]
30	20, 29	leq 91	. . . 4 |- [y:(al -> *) = \p:al T.] |= [\z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)] = \z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> T.]]
31	16, 21, 30	ceq2 90	. . 3 |- [y:(al -> *) = \p:al T.] |= [(A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)]) = (A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> T.])]
32	22, 23, 31	cla4ev 169	. 2 |- (A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> T.]) |= (E.\y:(al -> *) (A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)]))
33	15, 32	syl 16	1 |- T. |= (E.\y:(al -> *) (A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z: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  kct 10   |= wffMMJ2 11  wffMMJ2t 12   /\ tan 119   ==> 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)