Theorem axext 219

Description: Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461. (Contributed by Mario Carneiro, 10-Oct-2014.)

Hypotheses

Ref	Expression
axext.1	|- A:(al -> *)
axext.2	|- B:(al -> *)

Assertion

Ref	Expression
axext	|- T. |= [(A.\x:al [(Ax:al) = (Bx:al)]) ==> [A = B]]

Distinct variable groups:   x,A   x,B   al,x

Proof of Theorem axext

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axext.1	. . . . . 6 |- A:(al -> *)
2		wv 64	. . . . . 6 |- x:al:al
3	1, 2	wc 50	. . . . 5 |- (Ax:al):*
4	3	wl 66	. . . 4 |- \x:al (Ax:al):(al -> *)
5		axext.2	. . . . . . . 8 |- B:(al -> *)
6	5, 2	wc 50	. . . . . . 7 |- (Bx:al):*
7	3, 6	weqi 76	. . . . . 6 |- [(Ax:al) = (Bx:al)]:*
8	7	ax4 150	. . . . 5 |- (A.\x:al [(Ax:al) = (Bx:al)]) |= [(Ax:al) = (Bx:al)]
9		wal 134	. . . . . 6 |- A.:((al -> *) -> *)
10	7	wl 66	. . . . . 6 |- \x:al [(Ax:al) = (Bx:al)]:(al -> *)
11		wv 64	. . . . . 6 |- y:al:al
12	9, 11	ax-17 105	. . . . . 6 |- T. |= [(\x:al A.y:al) = A.]
13	7, 11	ax-hbl1 103	. . . . . 6 |- T. |= [(\x:al \x:al [(Ax:al) = (Bx:al)]y:al) = \x:al [(Ax:al) = (Bx:al)]]
14	9, 10, 11, 12, 13	hbc 110	. . . . 5 |- T. |= [(\x:al (A.\x:al [(Ax:al) = (Bx:al)])y:al) = (A.\x:al [(Ax:al) = (Bx:al)])]
15	3, 8, 14	leqf 181	. . . 4 |- (A.\x:al [(Ax:al) = (Bx:al)]) |= [\x:al (Ax:al) = \x:al (Bx:al)]
16	15	ax-cb1 29	. . . . 5 |- (A.\x:al [(Ax:al) = (Bx:al)]):*
17	1	eta 178	. . . . 5 |- T. |= [\x:al (Ax:al) = A]
18	16, 17	a1i 28	. . . 4 |- (A.\x:al [(Ax:al) = (Bx:al)]) |= [\x:al (Ax:al) = A]
19	5	eta 178	. . . . 5 |- T. |= [\x:al (Bx:al) = B]
20	16, 19	a1i 28	. . . 4 |- (A.\x:al [(Ax:al) = (Bx:al)]) |= [\x:al (Bx:al) = B]
21	4, 15, 18, 20	3eqtr3i 97	. . 3 |- (A.\x:al [(Ax:al) = (Bx:al)]) |= [A = B]
22		wtru 43	. . 3 |- T.:*
23	21, 22	adantl 56	. 2 |- (T., (A.\x:al [(Ax:al) = (Bx:al)])) |= [A = B]
24	23	ex 158	1 |- T. |= [(A.\x:al [(Ax:al) = (Bx:al)]) ==> [A = B]]

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

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  ax-eta 177

This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129

This theorem is referenced by: (None)