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Theorem grothprim 9512
Description: The Tarski-Grothendieck Axiom ax-groth 9501 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 9510 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))
2 3anass 1034 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦))) ↔ (𝑥𝑦 ∧ (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))))
3 dfss2 3556 . . . . . . . . . . . . 13 (𝑤𝑧 ↔ ∀𝑢(𝑢𝑤𝑢𝑧))
4 elin 3757 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦𝑣) ↔ (𝑤𝑦𝑤𝑣))
53, 4imbi12i 338 . . . . . . . . . . . 12 ((𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ (∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
65albii 1736 . . . . . . . . . . 11 (∀𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
76rexbii 3022 . . . . . . . . . 10 (∃𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∃𝑣𝑦𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
8 df-rex 2901 . . . . . . . . . 10 (∃𝑣𝑦𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)) ↔ ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
97, 8bitri 262 . . . . . . . . 9 (∃𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
109ralbii 2962 . . . . . . . 8 (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑧𝑦𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
11 df-ral 2900 . . . . . . . 8 (∀𝑧𝑦𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))) ↔ ∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))))
1210, 11bitri 262 . . . . . . 7 (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))))
13 dfss2 3556 . . . . . . . . . . 11 (𝑧𝑦 ↔ ∀𝑤(𝑤𝑧𝑤𝑦))
14 vex 3175 . . . . . . . . . . . . . . 15 𝑦 ∈ V
15 difexg 4730 . . . . . . . . . . . . . . 15 (𝑦 ∈ V → (𝑦𝑧) ∈ V)
1614, 15ax-mp 5 . . . . . . . . . . . . . 14 (𝑦𝑧) ∈ V
17 vex 3175 . . . . . . . . . . . . . 14 𝑧 ∈ V
18 incom 3766 . . . . . . . . . . . . . . 15 ((𝑦𝑧) ∩ 𝑧) = (𝑧 ∩ (𝑦𝑧))
19 disjdif 3991 . . . . . . . . . . . . . . 15 (𝑧 ∩ (𝑦𝑧)) = ∅
2018, 19eqtri 2631 . . . . . . . . . . . . . 14 ((𝑦𝑧) ∩ 𝑧) = ∅
2116, 17, 20brdom6disj 9212 . . . . . . . . . . . . 13 ((𝑦𝑧) ≼ 𝑧 ↔ ∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤))
2221orbi1i 540 . . . . . . . . . . . 12 (((𝑦𝑧) ≼ 𝑧𝑧𝑦) ↔ (∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
23 19.44v 1898 . . . . . . . . . . . 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 265 . . . . . . . . . . 11 (((𝑦𝑧) ≼ 𝑧𝑧𝑦) ↔ ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
2513, 24imbi12i 338 . . . . . . . . . 10 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ (∀𝑤(𝑤𝑧𝑤𝑦) → ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
26 19.35 1793 . . . . . . . . . 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 265 . . . . . . . . 9 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ ∃𝑤((𝑤𝑧𝑤𝑦) → ((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
28 grothprimlem 9511 . . . . . . . . . . . . . . . . . 18 ({𝑣, 𝑢} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))))
2928mobii 2480 . . . . . . . . . . . . . . . . 17 (∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∃*𝑢𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))))
30 mo2v 2464 . . . . . . . . . . . . . . . . 17 (∃*𝑢𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) ↔ ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
3129, 30bitri 262 . . . . . . . . . . . . . . . 16 (∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
3231ralbii 2962 . . . . . . . . . . . . . . 15 (∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∀𝑣𝑧𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
33 df-ral 2900 . . . . . . . . . . . . . . 15 (∀𝑣𝑧𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡) ↔ ∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)))
3432, 33bitri 262 . . . . . . . . . . . . . 14 (∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)))
35 df-ral 2900 . . . . . . . . . . . . . . 15 (∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∀𝑣(𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤))
36 eldif 3549 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝑦𝑧) ↔ (𝑣𝑦 ∧ ¬ 𝑣𝑧))
37 grothprimlem 9511 . . . . . . . . . . . . . . . . . . . 20 ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
3837rexbii 3022 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑢𝑧𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
39 df-rex 2901 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑧𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))) ↔ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))
4038, 39bitri 262 . . . . . . . . . . . . . . . . . 18 (∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))
4136, 40imbi12i 338 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ((𝑣𝑦 ∧ ¬ 𝑣𝑧) → ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))
42 pm5.6 948 . . . . . . . . . . . . . . . . 17 (((𝑣𝑦 ∧ ¬ 𝑣𝑧) → ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))) ↔ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4341, 42bitri 262 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4443albii 1736 . . . . . . . . . . . . . . 15 (∀𝑣(𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4535, 44bitri 262 . . . . . . . . . . . . . 14 (∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4634, 45anbi12i 728 . . . . . . . . . . . . 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 1785 . . . . . . . . . . . . 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 265 . . . . . . . . . . . 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 540 . . . . . . . . . . 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 324 . . . . . . . . . 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 1763 . . . . . . . . 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 262 . . . . . . . 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 1736 . . . . . . 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 728 . . . . . 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 1785 . . . . . 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 265 . . . . 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 725 . . . 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 262 . . 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 1763 . 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 218 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 194  wo 381  wa 382  w3a 1030  wal 1472  wex 1694  wcel 1976  ∃*wmo 2458  wral 2895  wrex 2896  Vcvv 3172  cdif 3536  cin 3538  wss 3539  c0 3873  {cpr 4126   class class class wbr 4577  cdom 7816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-reg 8357  ax-inf2 8398  ax-cc 9117  ax-ac2 9145  ax-groth 9501
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-oi 8275  df-card 8625  df-acn 8628  df-ac 8799  df-cda 8850
This theorem is referenced by: (None)
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