Axiom ax-ac2 10486

Description: In order to avoid uses of ax-reg 9615 for derivation of AC equivalents, we provide ax-ac2 10486, 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 10488. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1790 available. The derivation of ax-ac2 10486 from ax-ac 10482 is shown by Theorem axac2 10489, and the reverse derivation by axac 10490. Note that we use ax-reg 9615 to derive ax-ac 10482 from ax-ac2 10486, but not to derive ax-ac2 10486 from ax-ac 10482. (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 2100	. . . . . . 7 wff 𝑦 ∈ 𝑥
4		vz	. . . . . . . . 9 setvar 𝑧
5	4, 1	wel 2100	. . . . . . . 8 wff 𝑧 ∈ 𝑦
6		vv	. . . . . . . . . . 11 setvar 𝑣
7	6, 2	wel 2100	. . . . . . . . . 10 wff 𝑣 ∈ 𝑥
8	1, 6	weq 1959	. . . . . . . . . . 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 2100	. . . . . . . . 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 2100	. . . . . . . 8 wff 𝑧 ∈ 𝑥
17	6, 4	wel 2100	. . . . . . . . . 10 wff 𝑣 ∈ 𝑧
18	6, 1	wel 2100	. . . . . . . . . 10 wff 𝑣 ∈ 𝑦
19	17, 18	wa 395	. . . . . . . . 9 wff (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)
20		vu	. . . . . . . . . . . 12 setvar 𝑢
21	20, 4	wel 2100	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑧
22	20, 1	wel 2100	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑦
23	21, 22	wa 395	. . . . . . . . . 10 wff (𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦)
24	20, 6	weq 1959	. . . . . . . . . 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 846	. . . . 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 1532	. . . 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 1774	. . 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 1532	. 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 1774	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  10487