Theorem axext 206

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.

Hypotheses

Ref	Expression
axext.1	⊢ A:(α → ∗)
axext.2	⊢ B:(α → ∗)

Assertion

Ref	Expression
axext	⊢ ⊤⊧[(∀λx:α [(Ax:α) = (Bx:α)]) ⇒ [A = B]]

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

Proof of Theorem axext

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axext.1	. . . . . 6 ⊢ A:(α → ∗)
2		wv 58	. . . . . 6 ⊢ x:α:α
3	1, 2	wc 45	. . . . 5 ⊢ (Ax:α):∗
4	3	wl 59	. . . 4 ⊢ λx:α (Ax:α):(α → ∗)
5		axext.2	. . . . . . . 8 ⊢ B:(α → ∗)
6	5, 2	wc 45	. . . . . . 7 ⊢ (Bx:α):∗
7	3, 6	weqi 68	. . . . . 6 ⊢ [(Ax:α) = (Bx:α)]:∗
8	7	ax4 140	. . . . 5 ⊢ (∀λx:α [(Ax:α) = (Bx:α)])⊧[(Ax:α) = (Bx:α)]
9		wal 124	. . . . . 6 ⊢ ∀:((α → ∗) → ∗)
10	7	wl 59	. . . . . 6 ⊢ λx:α [(Ax:α) = (Bx:α)]:(α → ∗)
11		wv 58	. . . . . 6 ⊢ y:α:α
12	9, 11	ax-17 95	. . . . . 6 ⊢ ⊤⊧[(λx:α ∀y:α) = ∀]
13	7, 11	ax-hbl1 93	. . . . . 6 ⊢ ⊤⊧[(λx:α λx:α [(Ax:α) = (Bx:α)]y:α) = λx:α [(Ax:α) = (Bx:α)]]
14	9, 10, 11, 12, 13	hbc 100	. . . . 5 ⊢ ⊤⊧[(λx:α (∀λx:α [(Ax:α) = (Bx:α)])y:α) = (∀λx:α [(Ax:α) = (Bx:α)])]
15	3, 8, 14	leqf 169	. . . 4 ⊢ (∀λx:α [(Ax:α) = (Bx:α)])⊧[λx:α (Ax:α) = λx:α (Bx:α)]
16	15	ax-cb1 29	. . . . 5 ⊢ (∀λx:α [(Ax:α) = (Bx:α)]):∗
17	1	eta 166	. . . . 5 ⊢ ⊤⊧[λx:α (Ax:α) = A]
18	16, 17	a1i 28	. . . 4 ⊢ (∀λx:α [(Ax:α) = (Bx:α)])⊧[λx:α (Ax:α) = A]
19	5	eta 166	. . . . 5 ⊢ ⊤⊧[λx:α (Bx:α) = B]
20	16, 19	a1i 28	. . . 4 ⊢ (∀λx:α [(Ax:α) = (Bx:α)])⊧[λx:α (Bx:α) = B]
21	4, 15, 18, 20	3eqtr3i 87	. . 3 ⊢ (∀λx:α [(Ax:α) = (Bx:α)])⊧[A = B]
22		wtru 40	. . 3 ⊢ ⊤:∗
23	21, 22	adantl 51	. 2 ⊢ (⊤, (∀λx:α [(Ax:α) = (Bx:α)]))⊧[A = B]
24	23	ex 148	1 ⊢ ⊤⊧[(∀λx:α [(Ax:α) = (Bx:α)]) ⇒ [A = B]]

Syntax hints:  tv 1   → ht 2  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12   ⇒ tim 111  ∀tal 112

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  ax-eta 165

This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119

This theorem is referenced by: (None)