Axiom ax-ac2 10203

Description: In order to avoid uses of ax-reg 9312 for derivation of AC equivalents, we provide ax-ac2 10203, which is equivalent to the standard AC of textbooks. This appears to be the shortest known equivalent to the standard AC when expressed in terms of set theory primitives. It was found by Kurt Maes as Theorem ackm 10205. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1801 available. The derivation of ax-ac2 10203 from ax-ac 10199 is shown by Theorem axac2 10206, and the reverse derivation by axac 10207. Note that we use ax-reg 9312 to derive ax-ac 10199 from ax-ac2 10203, but not to derive ax-ac2 10203 from ax-ac 10199. (Contributed by NM, 19-Dec-2016.)

Assertion

Ref	Expression
ax-ac2	⊢ ∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))

Distinct variable group:   𝑥,𝑦,𝑧,𝑣,𝑢

Detailed syntax breakdown of Axiom ax-ac2

Step	Hyp	Ref	Expression
1		vy	. . . . . . . 8 setvar 𝑦
2		vx	. . . . . . . 8 setvar 𝑥
3	1, 2	wel 2110	. . . . . . 7 wff 𝑦 ∈ 𝑥
4		vz	. . . . . . . . 9 setvar 𝑧
5	4, 1	wel 2110	. . . . . . . 8 wff 𝑧 ∈ 𝑦
6		vv	. . . . . . . . . . 11 setvar 𝑣
7	6, 2	wel 2110	. . . . . . . . . 10 wff 𝑣 ∈ 𝑥
8	1, 6	weq 1969	. . . . . . . . . . 11 wff 𝑦 = 𝑣
9	8	wn 3	. . . . . . . . . 10 wff ¬ 𝑦 = 𝑣
10	7, 9	wa 395	. . . . . . . . 9 wff (𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣)
11	4, 6	wel 2110	. . . . . . . . 9 wff 𝑧 ∈ 𝑣
12	10, 11	wa 395	. . . . . . . 8 wff ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣)
13	5, 12	wi 4	. . . . . . 7 wff (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))
14	3, 13	wa 395	. . . . . 6 wff (𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣)))
15	3	wn 3	. . . . . . 7 wff ¬ 𝑦 ∈ 𝑥
16	4, 2	wel 2110	. . . . . . . 8 wff 𝑧 ∈ 𝑥
17	6, 4	wel 2110	. . . . . . . . . 10 wff 𝑣 ∈ 𝑧
18	6, 1	wel 2110	. . . . . . . . . 10 wff 𝑣 ∈ 𝑦
19	17, 18	wa 395	. . . . . . . . 9 wff (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)
20		vu	. . . . . . . . . . . 12 setvar 𝑢
21	20, 4	wel 2110	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑧
22	20, 1	wel 2110	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑦
23	21, 22	wa 395	. . . . . . . . . 10 wff (𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦)
24	20, 6	weq 1969	. . . . . . . . . 10 wff 𝑢 = 𝑣
25	23, 24	wi 4	. . . . . . . . 9 wff ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)
26	19, 25	wa 395	. . . . . . . 8 wff ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))
27	16, 26	wi 4	. . . . . . 7 wff (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))
28	15, 27	wa 395	. . . . . 6 wff (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
29	14, 28	wo 843	. . . . 5 wff ((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))
30	29, 20	wal 1539	. . . 4 wff ∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))
31	30, 6	wex 1785	. . 3 wff ∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))
32	31, 4	wal 1539	. 2 wff ∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))
33	32, 1	wex 1785	1 wff ∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))

This axiom is referenced by:  axac3  10204