Axiom ax-ac2 9600

Description: In order to avoid uses of ax-reg 8766 for derivation of AC equivalents, we provide ax-ac2 9600, 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 9602. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1896 available. The derivation of ax-ac2 9600 from ax-ac 9596 is shown by theorem axac2 9603, and the reverse derivation by axac 9604. Note that we use ax-reg 8766 to derive ax-ac 9596 from ax-ac2 9600, but not to derive ax-ac2 9600 from ax-ac 9596. (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 2167	. . . . . . 7 wff 𝑦 ∈ 𝑥
4		vz	. . . . . . . . 9 setvar 𝑧
5	4, 1	wel 2167	. . . . . . . 8 wff 𝑧 ∈ 𝑦
6		vv	. . . . . . . . . . 11 setvar 𝑣
7	6, 2	wel 2167	. . . . . . . . . 10 wff 𝑣 ∈ 𝑥
8	1, 6	weq 2063	. . . . . . . . . . 11 wff 𝑦 = 𝑣
9	8	wn 3	. . . . . . . . . 10 wff ¬ 𝑦 = 𝑣
10	7, 9	wa 386	. . . . . . . . 9 wff (𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣)
11	4, 6	wel 2167	. . . . . . . . 9 wff 𝑧 ∈ 𝑣
12	10, 11	wa 386	. . . . . . . 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 386	. . . . . 6 wff (𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣)))
15	3	wn 3	. . . . . . 7 wff ¬ 𝑦 ∈ 𝑥
16	4, 2	wel 2167	. . . . . . . 8 wff 𝑧 ∈ 𝑥
17	6, 4	wel 2167	. . . . . . . . . 10 wff 𝑣 ∈ 𝑧
18	6, 1	wel 2167	. . . . . . . . . 10 wff 𝑣 ∈ 𝑦
19	17, 18	wa 386	. . . . . . . . 9 wff (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)
20		vu	. . . . . . . . . . . 12 setvar 𝑢
21	20, 4	wel 2167	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑧
22	20, 1	wel 2167	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑦
23	21, 22	wa 386	. . . . . . . . . 10 wff (𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦)
24	20, 6	weq 2063	. . . . . . . . . 10 wff 𝑢 = 𝑣
25	23, 24	wi 4	. . . . . . . . 9 wff ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)
26	19, 25	wa 386	. . . . . . . 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 386	. . . . . 6 wff (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
29	14, 28	wo 880	. . . . 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 1656	. . . 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 1880	. . 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 1656	. 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 1880	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  9601