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Axiom ax-ac2 10102
Description: In order to avoid uses of ax-reg 9233 for derivation of AC equivalents, we provide ax-ac2 10102, 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 10104. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1803 available. The derivation of ax-ac2 10102 from ax-ac 10098 is shown by Theorem axac2 10105, and the reverse derivation by axac 10106. Note that we use ax-reg 9233 to derive ax-ac 10098 from ax-ac2 10102, but not to derive ax-ac2 10102 from ax-ac 10098. (Contributed by NM, 19-Dec-2016.)
Assertion
Ref Expression
ax-ac2 𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
Distinct variable group:   𝑥,𝑦,𝑧,𝑣,𝑢

Detailed syntax breakdown of Axiom ax-ac2
StepHypRef Expression
1 vy . . . . . . . 8 setvar 𝑦
2 vx . . . . . . . 8 setvar 𝑥
31, 2wel 2112 . . . . . . 7 wff 𝑦𝑥
4 vz . . . . . . . . 9 setvar 𝑧
54, 1wel 2112 . . . . . . . 8 wff 𝑧𝑦
6 vv . . . . . . . . . . 11 setvar 𝑣
76, 2wel 2112 . . . . . . . . . 10 wff 𝑣𝑥
81, 6weq 1971 . . . . . . . . . . 11 wff 𝑦 = 𝑣
98wn 3 . . . . . . . . . 10 wff ¬ 𝑦 = 𝑣
107, 9wa 399 . . . . . . . . 9 wff (𝑣𝑥 ∧ ¬ 𝑦 = 𝑣)
114, 6wel 2112 . . . . . . . . 9 wff 𝑧𝑣
1210, 11wa 399 . . . . . . . 8 wff ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣)
135, 12wi 4 . . . . . . 7 wff (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))
143, 13wa 399 . . . . . 6 wff (𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣)))
153wn 3 . . . . . . 7 wff ¬ 𝑦𝑥
164, 2wel 2112 . . . . . . . 8 wff 𝑧𝑥
176, 4wel 2112 . . . . . . . . . 10 wff 𝑣𝑧
186, 1wel 2112 . . . . . . . . . 10 wff 𝑣𝑦
1917, 18wa 399 . . . . . . . . 9 wff (𝑣𝑧𝑣𝑦)
20 vu . . . . . . . . . . . 12 setvar 𝑢
2120, 4wel 2112 . . . . . . . . . . 11 wff 𝑢𝑧
2220, 1wel 2112 . . . . . . . . . . 11 wff 𝑢𝑦
2321, 22wa 399 . . . . . . . . . 10 wff (𝑢𝑧𝑢𝑦)
2420, 6weq 1971 . . . . . . . . . 10 wff 𝑢 = 𝑣
2523, 24wi 4 . . . . . . . . 9 wff ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)
2619, 25wa 399 . . . . . . . 8 wff ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))
2716, 26wi 4 . . . . . . 7 wff (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))
2815, 27wa 399 . . . . . 6 wff 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))
2914, 28wo 847 . . . . 5 wff ((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
3029, 20wal 1541 . . . 4 wff 𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
3130, 6wex 1787 . . 3 wff 𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
3231, 4wal 1541 . 2 wff 𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
3332, 1wex 1787 1 wff 𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
Colors of variables: wff setvar class
This axiom is referenced by:  axac3  10103
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