Theorem ac 184

Description: Defining property of the indefinite descriptor: it selects an element from any type. This is equivalent to global choice in ZF.

Hypotheses

Ref	Expression
ac.1	⊢ F:(α → ∗)
ac.2	⊢ A:α

Assertion

Ref	Expression
ac	⊢ (FA)⊧(F(εF))

Proof of Theorem ac

Dummy variables x p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ac.1	. . 3 ⊢ F:(α → ∗)
2		wat 180	. . . 4 ⊢ ε:((α → ∗) → α)
3	2, 1	wc 45	. . 3 ⊢ (εF):α
4	1, 3	wc 45	. 2 ⊢ (F(εF)):∗
5		ac.2	. . . 4 ⊢ A:α
6	1, 5	wc 45	. . 3 ⊢ (FA):∗
7	6	id 25	. 2 ⊢ (FA)⊧(FA)
8	7	ax-cb1 29	. . 3 ⊢ (FA):∗
9		ax-ac 183	. . . . 5 ⊢ ⊤⊧(∀λp:(α → ∗) (∀λx:α [(p:(α → ∗)x:α) ⇒ (p:(α → ∗)(εp:(α → ∗)))]))
10		wal 124	. . . . . . 7 ⊢ ∀:((α → ∗) → ∗)
11		wim 127	. . . . . . . . 9 ⊢ ⇒ :(∗ → (∗ → ∗))
12		wv 58	. . . . . . . . . 10 ⊢ p:(α → ∗):(α → ∗)
13		wv 58	. . . . . . . . . 10 ⊢ x:α:α
14	12, 13	wc 45	. . . . . . . . 9 ⊢ (p:(α → ∗)x:α):∗
15	2, 12	wc 45	. . . . . . . . . 10 ⊢ (εp:(α → ∗)):α
16	12, 15	wc 45	. . . . . . . . 9 ⊢ (p:(α → ∗)(εp:(α → ∗))):∗
17	11, 14, 16	wov 64	. . . . . . . 8 ⊢ [(p:(α → ∗)x:α) ⇒ (p:(α → ∗)(εp:(α → ∗)))]:∗
18	17	wl 59	. . . . . . 7 ⊢ λx:α [(p:(α → ∗)x:α) ⇒ (p:(α → ∗)(εp:(α → ∗)))]:(α → ∗)
19	10, 18	wc 45	. . . . . 6 ⊢ (∀λx:α [(p:(α → ∗)x:α) ⇒ (p:(α → ∗)(εp:(α → ∗)))]):∗
20	12, 1	weqi 68	. . . . . . . . . . 11 ⊢ [p:(α → ∗) = F]:∗
21	20	id 25	. . . . . . . . . 10 ⊢ [p:(α → ∗) = F]⊧[p:(α → ∗) = F]
22	12, 13, 21	ceq1 79	. . . . . . . . 9 ⊢ [p:(α → ∗) = F]⊧[(p:(α → ∗)x:α) = (Fx:α)]
23	2, 12, 21	ceq2 80	. . . . . . . . . 10 ⊢ [p:(α → ∗) = F]⊧[(εp:(α → ∗)) = (εF)]
24	12, 15, 21, 23	ceq12 78	. . . . . . . . 9 ⊢ [p:(α → ∗) = F]⊧[(p:(α → ∗)(εp:(α → ∗))) = (F(εF))]
25	11, 14, 16, 22, 24	oveq12 90	. . . . . . . 8 ⊢ [p:(α → ∗) = F]⊧[[(p:(α → ∗)x:α) ⇒ (p:(α → ∗)(εp:(α → ∗)))] = [(Fx:α) ⇒ (F(εF))]]
26	17, 25	leq 81	. . . . . . 7 ⊢ [p:(α → ∗) = F]⊧[λx:α [(p:(α → ∗)x:α) ⇒ (p:(α → ∗)(εp:(α → ∗)))] = λx:α [(Fx:α) ⇒ (F(εF))]]
27	10, 18, 26	ceq2 80	. . . . . 6 ⊢ [p:(α → ∗) = F]⊧[(∀λx:α [(p:(α → ∗)x:α) ⇒ (p:(α → ∗)(εp:(α → ∗)))]) = (∀λx:α [(Fx:α) ⇒ (F(εF))])]
28	19, 1, 27	cla4v 142	. . . . 5 ⊢ (∀λp:(α → ∗) (∀λx:α [(p:(α → ∗)x:α) ⇒ (p:(α → ∗)(εp:(α → ∗)))]))⊧(∀λx:α [(Fx:α) ⇒ (F(εF))])
29	9, 28	syl 16	. . . 4 ⊢ ⊤⊧(∀λx:α [(Fx:α) ⇒ (F(εF))])
30	17, 25	eqtypi 69	. . . . 5 ⊢ [(Fx:α) ⇒ (F(εF))]:∗
31	1, 13	wc 45	. . . . . 6 ⊢ (Fx:α):∗
32	13, 5	weqi 68	. . . . . . . 8 ⊢ [x:α = A]:∗
33	32	id 25	. . . . . . 7 ⊢ [x:α = A]⊧[x:α = A]
34	1, 13, 33	ceq2 80	. . . . . 6 ⊢ [x:α = A]⊧[(Fx:α) = (FA)]
35	11, 31, 4, 34	oveq1 89	. . . . 5 ⊢ [x:α = A]⊧[[(Fx:α) ⇒ (F(εF))] = [(FA) ⇒ (F(εF))]]
36	30, 5, 35	cla4v 142	. . . 4 ⊢ (∀λx:α [(Fx:α) ⇒ (F(εF))])⊧[(FA) ⇒ (F(εF))]
37	29, 36	syl 16	. . 3 ⊢ ⊤⊧[(FA) ⇒ (F(εF))]
38	8, 37	a1i 28	. 2 ⊢ (FA)⊧[(FA) ⇒ (F(εF))]
39	4, 7, 38	mpd 146	1 ⊢ (FA)⊧(F(εF))

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  εtat 179

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-ac 183

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

This theorem is referenced by:  dfex2  185  exmid  186