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Axiom ax-groth 9333
Description: The Tarski-Grothendieck Axiom. For every set 𝑥 there is an inaccessible cardinal 𝑦 such that 𝑦 is not in 𝑥. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics." This version of the axiom is used by the Mizar project (http://www.mizar.org/JFM/Axiomatics/tarski.html). Unlike the ZFC axioms, this axiom is very long when expressed in terms of primitive symbols - see grothprim 9344. An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.)
Assertion
Ref Expression
ax-groth 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣

Detailed syntax breakdown of Axiom ax-groth
StepHypRef Expression
1 vx . . . 4 setvar 𝑥
2 vy . . . 4 setvar 𝑦
31, 2wel 1938 . . 3 wff 𝑥𝑦
4 vw . . . . . . . . 9 setvar 𝑤
54cv 1467 . . . . . . . 8 class 𝑤
6 vz . . . . . . . . 9 setvar 𝑧
76cv 1467 . . . . . . . 8 class 𝑧
85, 7wss 3426 . . . . . . 7 wff 𝑤𝑧
94, 2wel 1938 . . . . . . 7 wff 𝑤𝑦
108, 9wi 4 . . . . . 6 wff (𝑤𝑧𝑤𝑦)
1110, 4wal 1466 . . . . 5 wff 𝑤(𝑤𝑧𝑤𝑦)
12 vv . . . . . . . . . 10 setvar 𝑣
1312cv 1467 . . . . . . . . 9 class 𝑣
1413, 7wss 3426 . . . . . . . 8 wff 𝑣𝑧
1512, 4wel 1938 . . . . . . . 8 wff 𝑣𝑤
1614, 15wi 4 . . . . . . 7 wff (𝑣𝑧𝑣𝑤)
1716, 12wal 1466 . . . . . 6 wff 𝑣(𝑣𝑧𝑣𝑤)
182cv 1467 . . . . . 6 class 𝑦
1917, 4, 18wrex 2792 . . . . 5 wff 𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)
2011, 19wa 378 . . . 4 wff (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤))
2120, 6, 18wral 2791 . . 3 wff 𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤))
227, 18wss 3426 . . . . 5 wff 𝑧𝑦
23 cen 7649 . . . . . . 7 class
247, 18, 23wbr 4434 . . . . . 6 wff 𝑧𝑦
256, 2wel 1938 . . . . . 6 wff 𝑧𝑦
2624, 25wo 377 . . . . 5 wff (𝑧𝑦𝑧𝑦)
2722, 26wi 4 . . . 4 wff (𝑧𝑦 → (𝑧𝑦𝑧𝑦))
2827, 6wal 1466 . . 3 wff 𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦))
293, 21, 28w3a 1021 . 2 wff (𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
3029, 2wex 1692 1 wff 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
Colors of variables: wff setvar class
This axiom is referenced by:  axgroth5  9334  axgroth2  9335
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