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Theorem grothprim 10248
Description: The Tarski-Grothendieck Axiom ax-groth 10237 expanded into set theory primitives using 163 symbols (allowing the defined symbols , , , and ). An open problem is whether a shorter equivalent exists (when expanded to primitives). (Contributed by NM, 16-Apr-2007.)
Assertion
Ref Expression
grothprim 𝑦(𝑥𝑦 ∧ ∀𝑧((𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦))))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,,𝑔

Proof of Theorem grothprim
StepHypRef Expression
1 axgroth4 10246 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))
2 3anass 1089 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦))) ↔ (𝑥𝑦 ∧ (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))))
3 dfss2 3953 . . . . . . . . . . . . 13 (𝑤𝑧 ↔ ∀𝑢(𝑢𝑤𝑢𝑧))
4 elin 4167 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦𝑣) ↔ (𝑤𝑦𝑤𝑣))
53, 4imbi12i 353 . . . . . . . . . . . 12 ((𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ (∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
65albii 1813 . . . . . . . . . . 11 (∀𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
76rexbii 3245 . . . . . . . . . 10 (∃𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∃𝑣𝑦𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
8 df-rex 3142 . . . . . . . . . 10 (∃𝑣𝑦𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)) ↔ ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
97, 8bitri 277 . . . . . . . . 9 (∃𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
109ralbii 3163 . . . . . . . 8 (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑧𝑦𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
11 df-ral 3141 . . . . . . . 8 (∀𝑧𝑦𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))) ↔ ∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))))
1210, 11bitri 277 . . . . . . 7 (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))))
13 dfss2 3953 . . . . . . . . . . 11 (𝑧𝑦 ↔ ∀𝑤(𝑤𝑧𝑤𝑦))
14 vex 3496 . . . . . . . . . . . . . . 15 𝑦 ∈ V
1514difexi 5223 . . . . . . . . . . . . . 14 (𝑦𝑧) ∈ V
16 vex 3496 . . . . . . . . . . . . . 14 𝑧 ∈ V
17 incom 4176 . . . . . . . . . . . . . . 15 ((𝑦𝑧) ∩ 𝑧) = (𝑧 ∩ (𝑦𝑧))
18 disjdif 4419 . . . . . . . . . . . . . . 15 (𝑧 ∩ (𝑦𝑧)) = ∅
1917, 18eqtri 2842 . . . . . . . . . . . . . 14 ((𝑦𝑧) ∩ 𝑧) = ∅
2015, 16, 19brdom6disj 9946 . . . . . . . . . . . . 13 ((𝑦𝑧) ≼ 𝑧 ↔ ∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤))
2120orbi1i 909 . . . . . . . . . . . 12 (((𝑦𝑧) ≼ 𝑧𝑧𝑦) ↔ (∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
22 19.44v 1992 . . . . . . . . . . . 12 (∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦) ↔ (∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
2321, 22bitr4i 280 . . . . . . . . . . 11 (((𝑦𝑧) ≼ 𝑧𝑧𝑦) ↔ ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
2413, 23imbi12i 353 . . . . . . . . . 10 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ (∀𝑤(𝑤𝑧𝑤𝑦) → ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
25 19.35 1871 . . . . . . . . . 10 (∃𝑤((𝑤𝑧𝑤𝑦) → ((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)) ↔ (∀𝑤(𝑤𝑧𝑤𝑦) → ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
2624, 25bitr4i 280 . . . . . . . . 9 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ ∃𝑤((𝑤𝑧𝑤𝑦) → ((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
27 grothprimlem 10247 . . . . . . . . . . . . . . . . . 18 ({𝑣, 𝑢} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))))
2827mobii 2625 . . . . . . . . . . . . . . . . 17 (∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∃*𝑢𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))))
29 df-mo 2616 . . . . . . . . . . . . . . . . 17 (∃*𝑢𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) ↔ ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
3028, 29bitri 277 . . . . . . . . . . . . . . . 16 (∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
3130ralbii 3163 . . . . . . . . . . . . . . 15 (∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∀𝑣𝑧𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
32 df-ral 3141 . . . . . . . . . . . . . . 15 (∀𝑣𝑧𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡) ↔ ∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)))
3331, 32bitri 277 . . . . . . . . . . . . . 14 (∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)))
34 df-ral 3141 . . . . . . . . . . . . . . 15 (∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∀𝑣(𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤))
35 eldif 3944 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝑦𝑧) ↔ (𝑣𝑦 ∧ ¬ 𝑣𝑧))
36 grothprimlem 10247 . . . . . . . . . . . . . . . . . . . 20 ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
3736rexbii 3245 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑢𝑧𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
38 df-rex 3142 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑧𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))) ↔ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))
3937, 38bitri 277 . . . . . . . . . . . . . . . . . 18 (∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))
4035, 39imbi12i 353 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ((𝑣𝑦 ∧ ¬ 𝑣𝑧) → ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))
41 pm5.6 997 . . . . . . . . . . . . . . . . 17 (((𝑣𝑦 ∧ ¬ 𝑣𝑧) → ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))) ↔ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4240, 41bitri 277 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4342albii 1813 . . . . . . . . . . . . . . 15 (∀𝑣(𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4434, 43bitri 277 . . . . . . . . . . . . . 14 (∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4533, 44anbi12i 628 . . . . . . . . . . . . 13 ((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ (∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))))
46 19.26 1864 . . . . . . . . . . . . 13 (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ↔ (∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))))
4745, 46bitr4i 280 . . . . . . . . . . . 12 ((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))))
4847orbi1i 909 . . . . . . . . . . 11 (((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦) ↔ (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦))
4948imbi2i 338 . . . . . . . . . 10 (((𝑤𝑧𝑤𝑦) → ((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)) ↔ ((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦)))
5049exbii 1841 . . . . . . . . 9 (∃𝑤((𝑤𝑧𝑤𝑦) → ((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)) ↔ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦)))
5126, 50bitri 277 . . . . . . . 8 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦)))
5251albii 1813 . . . . . . 7 (∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ ∀𝑧𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦)))
5312, 52anbi12i 628 . . . . . 6 ((∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦))) ↔ (∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∀𝑧𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦))))
54 19.26 1864 . . . . . 6 (∀𝑧((𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦))) ↔ (∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∀𝑧𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦))))
5553, 54bitr4i 280 . . . . 5 ((∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦))) ↔ ∀𝑧((𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦))))
5655anbi2i 624 . . . 4 ((𝑥𝑦 ∧ (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))) ↔ (𝑥𝑦 ∧ ∀𝑧((𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦)))))
572, 56bitri 277 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦))) ↔ (𝑥𝑦 ∧ ∀𝑧((𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦)))))
5857exbii 1841 . 2 (∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦))) ↔ ∃𝑦(𝑥𝑦 ∧ ∀𝑧((𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦)))))
591, 58mpbi 232 1 𝑦(𝑥𝑦 ∧ ∀𝑧((𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3a 1081  wal 1528  wex 1773  wcel 2107  ∃*wmo 2614  wral 3136  wrex 3137  cdif 3931  cin 3933  wss 3934  c0 4289  {cpr 4561   class class class wbr 5057  cdom 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-reg 9048  ax-inf2 9096  ax-cc 9849  ax-ac2 9877  ax-groth 10237
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-oi 8966  df-dju 9322  df-card 9360  df-acn 9363  df-ac 9534
This theorem is referenced by: (None)
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