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Axiom ax-ac2 9877
Description: In order to avoid uses of ax-reg 9048 for derivation of AC equivalents, we provide ax-ac2 9877, 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 9879. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1789 available. The derivation of ax-ac2 9877 from ax-ac 9873 is shown by theorem axac2 9880, and the reverse derivation by axac 9881. Note that we use ax-reg 9048 to derive ax-ac 9873 from ax-ac2 9877, but not to derive ax-ac2 9877 from ax-ac 9873. (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 2107 . . . . . . 7 wff 𝑦𝑥
4 vz . . . . . . . . 9 setvar 𝑧
54, 1wel 2107 . . . . . . . 8 wff 𝑧𝑦
6 vv . . . . . . . . . . 11 setvar 𝑣
76, 2wel 2107 . . . . . . . . . 10 wff 𝑣𝑥
81, 6weq 1957 . . . . . . . . . . 11 wff 𝑦 = 𝑣
98wn 3 . . . . . . . . . 10 wff ¬ 𝑦 = 𝑣
107, 9wa 396 . . . . . . . . 9 wff (𝑣𝑥 ∧ ¬ 𝑦 = 𝑣)
114, 6wel 2107 . . . . . . . . 9 wff 𝑧𝑣
1210, 11wa 396 . . . . . . . 8 wff ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣)
135, 12wi 4 . . . . . . 7 wff (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))
143, 13wa 396 . . . . . 6 wff (𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣)))
153wn 3 . . . . . . 7 wff ¬ 𝑦𝑥
164, 2wel 2107 . . . . . . . 8 wff 𝑧𝑥
176, 4wel 2107 . . . . . . . . . 10 wff 𝑣𝑧
186, 1wel 2107 . . . . . . . . . 10 wff 𝑣𝑦
1917, 18wa 396 . . . . . . . . 9 wff (𝑣𝑧𝑣𝑦)
20 vu . . . . . . . . . . . 12 setvar 𝑢
2120, 4wel 2107 . . . . . . . . . . 11 wff 𝑢𝑧
2220, 1wel 2107 . . . . . . . . . . 11 wff 𝑢𝑦
2321, 22wa 396 . . . . . . . . . 10 wff (𝑢𝑧𝑢𝑦)
2420, 6weq 1957 . . . . . . . . . 10 wff 𝑢 = 𝑣
2523, 24wi 4 . . . . . . . . 9 wff ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)
2619, 25wa 396 . . . . . . . 8 wff ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))
2716, 26wi 4 . . . . . . 7 wff (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))
2815, 27wa 396 . . . . . 6 wff 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))
2914, 28wo 843 . . . . 5 wff ((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
3029, 20wal 1528 . . . 4 wff 𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
3130, 6wex 1773 . . 3 wff 𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
3231, 4wal 1528 . 2 wff 𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
3332, 1wex 1773 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  9878
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