Axiom ax-ac2 9285

Description: In order to avoid uses of ax-reg 8497 for derivation of AC equivalents, we provide ax-ac2 9285, 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 9287. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1722 available. The derivation of ax-ac2 9285 from ax-ac 9281 is shown by theorem axac2 9288, and the reverse derivation by axac 9289. Note that we use ax-reg 8497 to derive ax-ac 9281 from ax-ac2 9285, but not to derive ax-ac2 9285 from ax-ac 9281. (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 1991	. . . . . . 7 wff 𝑦 ∈ 𝑥
4		vz	. . . . . . . . 9 setvar 𝑧
5	4, 1	wel 1991	. . . . . . . 8 wff 𝑧 ∈ 𝑦
6		vv	. . . . . . . . . . 11 setvar 𝑣
7	6, 2	wel 1991	. . . . . . . . . 10 wff 𝑣 ∈ 𝑥
8	1, 6	weq 1874	. . . . . . . . . . 11 wff 𝑦 = 𝑣
9	8	wn 3	. . . . . . . . . 10 wff ¬ 𝑦 = 𝑣
10	7, 9	wa 384	. . . . . . . . 9 wff (𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣)
11	4, 6	wel 1991	. . . . . . . . 9 wff 𝑧 ∈ 𝑣
12	10, 11	wa 384	. . . . . . . 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 384	. . . . . 6 wff (𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣)))
15	3	wn 3	. . . . . . 7 wff ¬ 𝑦 ∈ 𝑥
16	4, 2	wel 1991	. . . . . . . 8 wff 𝑧 ∈ 𝑥
17	6, 4	wel 1991	. . . . . . . . . 10 wff 𝑣 ∈ 𝑧
18	6, 1	wel 1991	. . . . . . . . . 10 wff 𝑣 ∈ 𝑦
19	17, 18	wa 384	. . . . . . . . 9 wff (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)
20		vu	. . . . . . . . . . . 12 setvar 𝑢
21	20, 4	wel 1991	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑧
22	20, 1	wel 1991	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑦
23	21, 22	wa 384	. . . . . . . . . 10 wff (𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦)
24	20, 6	weq 1874	. . . . . . . . . 10 wff 𝑢 = 𝑣
25	23, 24	wi 4	. . . . . . . . 9 wff ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)
26	19, 25	wa 384	. . . . . . . 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 384	. . . . . 6 wff (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
29	14, 28	wo 383	. . . . 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 1481	. . . 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 1704	. . 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 1481	. 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 1704	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  9286