Description: If M is a transitive class, then the following are equivalent. (1) Every nonempty set x e. M of pairwise disjoint nonempty sets has a choice set in M . (2) The class M models the Axiom of Choice, in the form ac8prim .
Lemma II.2.11(7) of Kunen2 p. 114. Kunen has the additional hypotheses that the Extensionality, Separation, Pairing, and Union axioms are true in M . This, apparently, is because Kunen's statement of the Axiom of Choice uses defined notions, including (/) and i^i , and these axioms guarantee that these notions are well-defined. When we state the axiom using primitives only, the need for these hypotheses disappears. (Contributed by Eric Schmidt, 19-Oct-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | modelac8prim | ⊢ ( Tr 𝑀 → ( ∀ 𝑥 ∈ 𝑀 ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) → ∃ 𝑦 ∈ 𝑀 ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝑀 ( ( ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 𝑤 ∈ 𝑧 ) ∧ ∀ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) → ∃ 𝑦 ∈ 𝑀 ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 ∀ 𝑣 ∈ 𝑀 ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ↔ 𝑣 = 𝑤 ) ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralabso | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → 𝑧 ≠ ∅ ) ) ) | |
| 2 | n0abso | ⊢ ( ( Tr 𝑀 ∧ 𝑧 ∈ 𝑀 ) → ( 𝑧 ≠ ∅ ↔ ∃ 𝑤 ∈ 𝑀 𝑤 ∈ 𝑧 ) ) | |
| 3 | 2 | adantlr | ⊢ ( ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑧 ∈ 𝑀 ) → ( 𝑧 ≠ ∅ ↔ ∃ 𝑤 ∈ 𝑀 𝑤 ∈ 𝑧 ) ) |
| 4 | 3 | imbi2d | ⊢ ( ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑧 ∈ 𝑀 ) → ( ( 𝑧 ∈ 𝑥 → 𝑧 ≠ ∅ ) ↔ ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 𝑤 ∈ 𝑧 ) ) ) |
| 5 | 4 | ralbidva | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → ( ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → 𝑧 ≠ ∅ ) ↔ ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 𝑤 ∈ 𝑧 ) ) ) |
| 6 | 1 5 | bitrd | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 𝑤 ∈ 𝑧 ) ) ) |
| 7 | simpl | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → Tr 𝑀 ) | |
| 8 | ralabso | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → ( ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ↔ ∀ 𝑤 ∈ 𝑀 ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ) | |
| 9 | 7 8 | ralabsobidv | ⊢ ( ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑥 ∈ 𝑀 ) → ( ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ↔ ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∀ 𝑤 ∈ 𝑀 ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ) ) |
| 10 | 9 | anabss3 | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → ( ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ↔ ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∀ 𝑤 ∈ 𝑀 ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ) ) |
| 11 | r19.21v | ⊢ ( ∀ 𝑤 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ↔ ( 𝑧 ∈ 𝑥 → ∀ 𝑤 ∈ 𝑀 ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ) | |
| 12 | impexp | ⊢ ( ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ↔ ( 𝑧 ∈ 𝑥 → ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ) | |
| 13 | df-ne | ⊢ ( 𝑧 ≠ 𝑤 ↔ ¬ 𝑧 = 𝑤 ) | |
| 14 | 13 | imbi1i | ⊢ ( ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ↔ ( ¬ 𝑧 = 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) |
| 15 | disjabso | ⊢ ( ( Tr 𝑀 ∧ 𝑧 ∈ 𝑀 ) → ( ( 𝑧 ∩ 𝑤 ) = ∅ ↔ ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) | |
| 16 | 15 | imbi2d | ⊢ ( ( Tr 𝑀 ∧ 𝑧 ∈ 𝑀 ) → ( ( ¬ 𝑧 = 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ↔ ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) |
| 17 | 14 16 | bitrid | ⊢ ( ( Tr 𝑀 ∧ 𝑧 ∈ 𝑀 ) → ( ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ↔ ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) |
| 18 | 17 | imbi2d | ⊢ ( ( Tr 𝑀 ∧ 𝑧 ∈ 𝑀 ) → ( ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ↔ ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) ) |
| 19 | 12 18 | bitr3id | ⊢ ( ( Tr 𝑀 ∧ 𝑧 ∈ 𝑀 ) → ( ( 𝑧 ∈ 𝑥 → ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ↔ ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) ) |
| 20 | 19 | ralbidv | ⊢ ( ( Tr 𝑀 ∧ 𝑧 ∈ 𝑀 ) → ( ∀ 𝑤 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ↔ ∀ 𝑤 ∈ 𝑀 ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) ) |
| 21 | 11 20 | bitr3id | ⊢ ( ( Tr 𝑀 ∧ 𝑧 ∈ 𝑀 ) → ( ( 𝑧 ∈ 𝑥 → ∀ 𝑤 ∈ 𝑀 ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ↔ ∀ 𝑤 ∈ 𝑀 ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) ) |
| 22 | 21 | ralbidva | ⊢ ( Tr 𝑀 → ( ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∀ 𝑤 ∈ 𝑀 ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ↔ ∀ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) ) |
| 23 | 22 | adantr | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → ( ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∀ 𝑤 ∈ 𝑀 ( 𝑤 ∈ 𝑥 → ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ) ↔ ∀ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) ) |
| 24 | 10 23 | bitrd | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → ( ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ↔ ∀ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) ) |
| 25 | 6 24 | anbi12d | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) ↔ ( ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 𝑤 ∈ 𝑧 ) ∧ ∀ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) ) ) |
| 26 | simpl | ⊢ ( ( Tr 𝑀 ∧ 𝑦 ∈ 𝑀 ) → Tr 𝑀 ) | |
| 27 | elin | ⊢ ( 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ↔ ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ) | |
| 28 | 27 | eubii | ⊢ ( ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ↔ ∃! 𝑣 ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ) |
| 29 | trel | ⊢ ( Tr 𝑀 → ( ( 𝑣 ∈ 𝑦 ∧ 𝑦 ∈ 𝑀 ) → 𝑣 ∈ 𝑀 ) ) | |
| 30 | 29 | imp | ⊢ ( ( Tr 𝑀 ∧ ( 𝑣 ∈ 𝑦 ∧ 𝑦 ∈ 𝑀 ) ) → 𝑣 ∈ 𝑀 ) |
| 31 | 30 | anass1rs | ⊢ ( ( ( Tr 𝑀 ∧ 𝑦 ∈ 𝑀 ) ∧ 𝑣 ∈ 𝑦 ) → 𝑣 ∈ 𝑀 ) |
| 32 | 31 | adantrl | ⊢ ( ( ( Tr 𝑀 ∧ 𝑦 ∈ 𝑀 ) ∧ ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ) → 𝑣 ∈ 𝑀 ) |
| 33 | 32 | reueubd | ⊢ ( ( Tr 𝑀 ∧ 𝑦 ∈ 𝑀 ) → ( ∃! 𝑣 ∈ 𝑀 ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ↔ ∃! 𝑣 ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ) ) |
| 34 | 28 33 | bitr4id | ⊢ ( ( Tr 𝑀 ∧ 𝑦 ∈ 𝑀 ) → ( ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ↔ ∃! 𝑣 ∈ 𝑀 ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ) ) |
| 35 | reu6 | ⊢ ( ∃! 𝑣 ∈ 𝑀 ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ↔ ∃ 𝑤 ∈ 𝑀 ∀ 𝑣 ∈ 𝑀 ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ↔ 𝑣 = 𝑤 ) ) | |
| 36 | 34 35 | bitrdi | ⊢ ( ( Tr 𝑀 ∧ 𝑦 ∈ 𝑀 ) → ( ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ↔ ∃ 𝑤 ∈ 𝑀 ∀ 𝑣 ∈ 𝑀 ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ↔ 𝑣 = 𝑤 ) ) ) |
| 37 | 26 36 | ralabsobidv | ⊢ ( ( ( Tr 𝑀 ∧ 𝑦 ∈ 𝑀 ) ∧ 𝑥 ∈ 𝑀 ) → ( ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ↔ ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 ∀ 𝑣 ∈ 𝑀 ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ↔ 𝑣 = 𝑤 ) ) ) ) |
| 38 | 37 | an32s | ⊢ ( ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑦 ∈ 𝑀 ) → ( ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ↔ ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 ∀ 𝑣 ∈ 𝑀 ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ↔ 𝑣 = 𝑤 ) ) ) ) |
| 39 | 38 | rexbidva | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → ( ∃ 𝑦 ∈ 𝑀 ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ↔ ∃ 𝑦 ∈ 𝑀 ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 ∀ 𝑣 ∈ 𝑀 ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ↔ 𝑣 = 𝑤 ) ) ) ) |
| 40 | 25 39 | imbi12d | ⊢ ( ( Tr 𝑀 ∧ 𝑥 ∈ 𝑀 ) → ( ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) → ∃ 𝑦 ∈ 𝑀 ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ↔ ( ( ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 𝑤 ∈ 𝑧 ) ∧ ∀ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) → ∃ 𝑦 ∈ 𝑀 ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 ∀ 𝑣 ∈ 𝑀 ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ↔ 𝑣 = 𝑤 ) ) ) ) ) |
| 41 | 40 | ralbidva | ⊢ ( Tr 𝑀 → ( ∀ 𝑥 ∈ 𝑀 ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) → ∃ 𝑦 ∈ 𝑀 ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝑀 ( ( ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 𝑤 ∈ 𝑧 ) ∧ ∀ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ¬ 𝑧 = 𝑤 → ∀ 𝑦 ∈ 𝑀 ( 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤 ) ) ) ) → ∃ 𝑦 ∈ 𝑀 ∀ 𝑧 ∈ 𝑀 ( 𝑧 ∈ 𝑥 → ∃ 𝑤 ∈ 𝑀 ∀ 𝑣 ∈ 𝑀 ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ↔ 𝑣 = 𝑤 ) ) ) ) ) |