Axiom ax-ac2 10458

Description: In order to avoid uses of ax-reg 9587 for derivation of AC equivalents, we provide ax-ac2 10458, 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 10460. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1798 available. The derivation of ax-ac2 10458 from ax-ac 10454 is shown by Theorem axac2 10461, and the reverse derivation by axac 10462. Note that we use ax-reg 9587 to derive ax-ac 10454 from ax-ac2 10458, but not to derive ax-ac2 10458 from ax-ac 10454. (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 2108	. . . . . . 7 wff 𝑦 ∈ 𝑥
4		vz	. . . . . . . . 9 setvar 𝑧
5	4, 1	wel 2108	. . . . . . . 8 wff 𝑧 ∈ 𝑦
6		vv	. . . . . . . . . . 11 setvar 𝑣
7	6, 2	wel 2108	. . . . . . . . . 10 wff 𝑣 ∈ 𝑥
8	1, 6	weq 1967	. . . . . . . . . . 11 wff 𝑦 = 𝑣
9	8	wn 3	. . . . . . . . . 10 wff ¬ 𝑦 = 𝑣
10	7, 9	wa 397	. . . . . . . . 9 wff (𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣)
11	4, 6	wel 2108	. . . . . . . . 9 wff 𝑧 ∈ 𝑣
12	10, 11	wa 397	. . . . . . . 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 397	. . . . . 6 wff (𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣)))
15	3	wn 3	. . . . . . 7 wff ¬ 𝑦 ∈ 𝑥
16	4, 2	wel 2108	. . . . . . . 8 wff 𝑧 ∈ 𝑥
17	6, 4	wel 2108	. . . . . . . . . 10 wff 𝑣 ∈ 𝑧
18	6, 1	wel 2108	. . . . . . . . . 10 wff 𝑣 ∈ 𝑦
19	17, 18	wa 397	. . . . . . . . 9 wff (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)
20		vu	. . . . . . . . . . . 12 setvar 𝑢
21	20, 4	wel 2108	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑧
22	20, 1	wel 2108	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑦
23	21, 22	wa 397	. . . . . . . . . 10 wff (𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦)
24	20, 6	weq 1967	. . . . . . . . . 10 wff 𝑢 = 𝑣
25	23, 24	wi 4	. . . . . . . . 9 wff ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)
26	19, 25	wa 397	. . . . . . . 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 397	. . . . . 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 1540	. . . 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 1782	. . 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 1540	. 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 1782	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  10459