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Theorem axun 209
Description: Axiom of Union. An axiom of Zermelo-Fraenkel set theory.
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.
StepHypRef Expression
1 wtru 40 . . . . . 6 |- T.:*
2 wex 129 . . . . . . 7 |- E.:(((al -> *) -> *) -> *)
3 wan 126 . . . . . . . . 9 |- /\ :(* -> (* -> *))
4 wv 58 . . . . . . . . . 10 |- x:(al -> *):(al -> *)
5 wv 58 . . . . . . . . . 10 |- z:al:al
64, 5wc 45 . . . . . . . . 9 |- (x:(al -> *)z:al):*
7 axun.1 . . . . . . . . . 10 |- A:((al -> *) -> *)
87, 4wc 45 . . . . . . . . 9 |- (Ax:(al -> *)):*
93, 6, 8wov 64 . . . . . . . 8 |- [(x:(al -> *)z:al) /\ (Ax:(al -> *))]:*
109wl 59 . . . . . . 7 |- \x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]:((al -> *) -> *)
112, 10wc 45 . . . . . 6 |- (E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]):*
121, 11wct 44 . . . . 5 |- (T., (E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))])):*
1312trud 27 . . . 4 |- (T., (E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))])) |= T.
1413ex 148 . . 3 |- T. |= [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> T.]
1514alrimiv 141 . 2 |- T. |= (A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> T.])
16 wal 124 . . . 4 |- A.:((al -> *) -> *)
17 wim 127 . . . . . 6 |- ==> :(* -> (* -> *))
18 wv 58 . . . . . . 7 |- y:(al -> *):(al -> *)
1918, 5wc 45 . . . . . 6 |- (y:(al -> *)z:al):*
2017, 11, 19wov 64 . . . . 5 |- [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)]:*
2120wl 59 . . . 4 |- \z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)]:(al -> *)
2216, 21wc 45 . . 3 |- (A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z:al) /\ (Ax:(al -> *))]) ==> (y:(al -> *)z:al)]):*
231wl 59 . . 3 |- \p:al T.:(al -> *)
2418, 23weqi 68 . . . . . . . . 9 |- [y:(al -> *) = \p:al T.]:*
2524id 25 . . . . . . . 8 |- [y:(al -> *) = \p:al T.] |= [y:(al -> *) = \p:al T.]
2618, 5, 25ceq1 79 . . . . . . 7 |- [y:(al -> *) = \p:al T.] |= [(y:(al -> *)z:al) = (\p:al T.z:al)]
271, 5, 24a17i 96 . . . . . . 7 |- [y:(al -> *) = \p:al T.] |= [(\p:al T.z:al) = T.]
2819, 26, 27eqtri 85 . . . . . 6 |- [y:(al -> *) = \p:al T.] |= [(y:(al -> *)z:al) = T.]
2917, 11, 19, 28oveq2 91 . . . . 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.]]
3020, 29leq 81 . . . 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.]]
3116, 21, 30ceq2 80 . . 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.])]
3222, 23, 31cla4ev 159 . 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)]))
3315, 32syl 16 1 |- T. |= (E.\y:(al -> *) (A.\z:al [(E.\x:(al -> *) [(x:(al -> *)z: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  kct 10   |= wffMMJ2 11  wffMMJ2t 12   /\ tan 109   ==> 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
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