MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grothprim Structured version   Visualization version   GIF version

Theorem grothprim 9978
Description: The Tarski-Grothendieck Axiom ax-groth 9967 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 9976 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))
2 3anass 1120 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦))) ↔ (𝑥𝑦 ∧ (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))))
3 dfss2 3815 . . . . . . . . . . . . 13 (𝑤𝑧 ↔ ∀𝑢(𝑢𝑤𝑢𝑧))
4 elin 4025 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦𝑣) ↔ (𝑤𝑦𝑤𝑣))
53, 4imbi12i 342 . . . . . . . . . . . 12 ((𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ (∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
65albii 1918 . . . . . . . . . . 11 (∀𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
76rexbii 3251 . . . . . . . . . 10 (∃𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∃𝑣𝑦𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
8 df-rex 3123 . . . . . . . . . 10 (∃𝑣𝑦𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)) ↔ ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
97, 8bitri 267 . . . . . . . . 9 (∃𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
109ralbii 3189 . . . . . . . 8 (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑧𝑦𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
11 df-ral 3122 . . . . . . . 8 (∀𝑧𝑦𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))) ↔ ∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))))
1210, 11bitri 267 . . . . . . 7 (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))))
13 dfss2 3815 . . . . . . . . . . 11 (𝑧𝑦 ↔ ∀𝑤(𝑤𝑧𝑤𝑦))
14 vex 3417 . . . . . . . . . . . . . . 15 𝑦 ∈ V
15 difexg 5035 . . . . . . . . . . . . . . 15 (𝑦 ∈ V → (𝑦𝑧) ∈ V)
1614, 15ax-mp 5 . . . . . . . . . . . . . 14 (𝑦𝑧) ∈ V
17 vex 3417 . . . . . . . . . . . . . 14 𝑧 ∈ V
18 incom 4034 . . . . . . . . . . . . . . 15 ((𝑦𝑧) ∩ 𝑧) = (𝑧 ∩ (𝑦𝑧))
19 disjdif 4265 . . . . . . . . . . . . . . 15 (𝑧 ∩ (𝑦𝑧)) = ∅
2018, 19eqtri 2849 . . . . . . . . . . . . . 14 ((𝑦𝑧) ∩ 𝑧) = ∅
2116, 17, 20brdom6disj 9676 . . . . . . . . . . . . 13 ((𝑦𝑧) ≼ 𝑧 ↔ ∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤))
2221orbi1i 942 . . . . . . . . . . . 12 (((𝑦𝑧) ≼ 𝑧𝑧𝑦) ↔ (∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
23 19.44v 2098 . . . . . . . . . . . 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))
2422, 23bitr4i 270 . . . . . . . . . . 11 (((𝑦𝑧) ≼ 𝑧𝑧𝑦) ↔ ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
2513, 24imbi12i 342 . . . . . . . . . 10 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ (∀𝑤(𝑤𝑧𝑤𝑦) → ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
26 19.35 1980 . . . . . . . . . 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)))
2725, 26bitr4i 270 . . . . . . . . 9 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ ∃𝑤((𝑤𝑧𝑤𝑦) → ((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
28 grothprimlem 9977 . . . . . . . . . . . . . . . . . 18 ({𝑣, 𝑢} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))))
2928mobii 2616 . . . . . . . . . . . . . . . . 17 (∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∃*𝑢𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))))
30 df-mo 2605 . . . . . . . . . . . . . . . . 17 (∃*𝑢𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) ↔ ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
3129, 30bitri 267 . . . . . . . . . . . . . . . 16 (∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
3231ralbii 3189 . . . . . . . . . . . . . . 15 (∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∀𝑣𝑧𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
33 df-ral 3122 . . . . . . . . . . . . . . 15 (∀𝑣𝑧𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡) ↔ ∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)))
3432, 33bitri 267 . . . . . . . . . . . . . 14 (∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)))
35 df-ral 3122 . . . . . . . . . . . . . . 15 (∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∀𝑣(𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤))
36 eldif 3808 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝑦𝑧) ↔ (𝑣𝑦 ∧ ¬ 𝑣𝑧))
37 grothprimlem 9977 . . . . . . . . . . . . . . . . . . . 20 ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
3837rexbii 3251 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑢𝑧𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
39 df-rex 3123 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑧𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))) ↔ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))
4038, 39bitri 267 . . . . . . . . . . . . . . . . . 18 (∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))
4136, 40imbi12i 342 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ((𝑣𝑦 ∧ ¬ 𝑣𝑧) → ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))
42 pm5.6 1029 . . . . . . . . . . . . . . . . 17 (((𝑣𝑦 ∧ ¬ 𝑣𝑧) → ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))) ↔ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4341, 42bitri 267 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4443albii 1918 . . . . . . . . . . . . . . 15 (∀𝑣(𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4535, 44bitri 267 . . . . . . . . . . . . . 14 (∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4634, 45anbi12i 620 . . . . . . . . . . . . 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))))))))
47 19.26 1972 . . . . . . . . . . . . 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))))))))
4846, 47bitr4i 270 . . . . . . . . . . . 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))))))))
4948orbi1i 942 . . . . . . . . . . 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))
5049imbi2i 328 . . . . . . . . . 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)))
5150exbii 1947 . . . . . . . . 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)))
5227, 51bitri 267 . . . . . . . 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)))
5352albii 1918 . . . . . . 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)))
5412, 53anbi12i 620 . . . . . 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))))
55 19.26 1972 . . . . . 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))))
5654, 55bitr4i 270 . . . . 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))))
5756anbi2i 616 . . . 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)))))
582, 57bitri 267 . . 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)))))
5958exbii 1947 . 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)))))
601, 59mpbi 222 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 198  wa 386  wo 878  w3a 1111  wal 1654  wex 1878  wcel 2164  ∃*wmo 2603  wral 3117  wrex 3118  Vcvv 3414  cdif 3795  cin 3797  wss 3798  c0 4146  {cpr 4401   class class class wbr 4875  cdom 8226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-reg 8773  ax-inf2 8822  ax-cc 9579  ax-ac2 9607  ax-groth 9967
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-oi 8691  df-card 9085  df-acn 9088  df-ac 9259  df-cda 9312
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator