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Theorem grothprim 9840
Description: The Tarski-Grothendieck Axiom ax-groth 9829 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 9838 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))
2 3anass 1081 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦))) ↔ (𝑥𝑦 ∧ (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))))
3 dfss2 3724 . . . . . . . . . . . . 13 (𝑤𝑧 ↔ ∀𝑢(𝑢𝑤𝑢𝑧))
4 elin 3931 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦𝑣) ↔ (𝑤𝑦𝑤𝑣))
53, 4imbi12i 339 . . . . . . . . . . . 12 ((𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ (∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
65albii 1888 . . . . . . . . . . 11 (∀𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
76rexbii 3171 . . . . . . . . . 10 (∃𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∃𝑣𝑦𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
8 df-rex 3048 . . . . . . . . . 10 (∃𝑣𝑦𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)) ↔ ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
97, 8bitri 264 . . . . . . . . 9 (∃𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
109ralbii 3110 . . . . . . . 8 (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑧𝑦𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
11 df-ral 3047 . . . . . . . 8 (∀𝑧𝑦𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))) ↔ ∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))))
1210, 11bitri 264 . . . . . . 7 (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))))
13 dfss2 3724 . . . . . . . . . . 11 (𝑧𝑦 ↔ ∀𝑤(𝑤𝑧𝑤𝑦))
14 vex 3335 . . . . . . . . . . . . . . 15 𝑦 ∈ V
15 difexg 4952 . . . . . . . . . . . . . . 15 (𝑦 ∈ V → (𝑦𝑧) ∈ V)
1614, 15ax-mp 5 . . . . . . . . . . . . . 14 (𝑦𝑧) ∈ V
17 vex 3335 . . . . . . . . . . . . . 14 𝑧 ∈ V
18 incom 3940 . . . . . . . . . . . . . . 15 ((𝑦𝑧) ∩ 𝑧) = (𝑧 ∩ (𝑦𝑧))
19 disjdif 4176 . . . . . . . . . . . . . . 15 (𝑧 ∩ (𝑦𝑧)) = ∅
2018, 19eqtri 2774 . . . . . . . . . . . . . 14 ((𝑦𝑧) ∩ 𝑧) = ∅
2116, 17, 20brdom6disj 9538 . . . . . . . . . . . . 13 ((𝑦𝑧) ≼ 𝑧 ↔ ∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤))
2221orbi1i 543 . . . . . . . . . . . 12 (((𝑦𝑧) ≼ 𝑧𝑧𝑦) ↔ (∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
23 19.44v 2069 . . . . . . . . . . . 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 267 . . . . . . . . . . 11 (((𝑦𝑧) ≼ 𝑧𝑧𝑦) ↔ ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
2513, 24imbi12i 339 . . . . . . . . . 10 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ (∀𝑤(𝑤𝑧𝑤𝑦) → ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
26 19.35 1946 . . . . . . . . . 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 267 . . . . . . . . 9 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ ∃𝑤((𝑤𝑧𝑤𝑦) → ((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
28 grothprimlem 9839 . . . . . . . . . . . . . . . . . 18 ({𝑣, 𝑢} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))))
2928mobii 2622 . . . . . . . . . . . . . . . . 17 (∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∃*𝑢𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))))
30 mo2v 2606 . . . . . . . . . . . . . . . . 17 (∃*𝑢𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) ↔ ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
3129, 30bitri 264 . . . . . . . . . . . . . . . 16 (∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
3231ralbii 3110 . . . . . . . . . . . . . . 15 (∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∀𝑣𝑧𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
33 df-ral 3047 . . . . . . . . . . . . . . 15 (∀𝑣𝑧𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡) ↔ ∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)))
3432, 33bitri 264 . . . . . . . . . . . . . 14 (∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)))
35 df-ral 3047 . . . . . . . . . . . . . . 15 (∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∀𝑣(𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤))
36 eldif 3717 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝑦𝑧) ↔ (𝑣𝑦 ∧ ¬ 𝑣𝑧))
37 grothprimlem 9839 . . . . . . . . . . . . . . . . . . . 20 ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
3837rexbii 3171 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑢𝑧𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
39 df-rex 3048 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑧𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))) ↔ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))
4038, 39bitri 264 . . . . . . . . . . . . . . . . . 18 (∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))
4136, 40imbi12i 339 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ((𝑣𝑦 ∧ ¬ 𝑣𝑧) → ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))
42 pm5.6 989 . . . . . . . . . . . . . . . . 17 (((𝑣𝑦 ∧ ¬ 𝑣𝑧) → ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))) ↔ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4341, 42bitri 264 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4443albii 1888 . . . . . . . . . . . . . . 15 (∀𝑣(𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4535, 44bitri 264 . . . . . . . . . . . . . 14 (∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4634, 45anbi12i 735 . . . . . . . . . . . . 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 1939 . . . . . . . . . . . . 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 267 . . . . . . . . . . . 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 543 . . . . . . . . . . 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 325 . . . . . . . . . 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 1915 . . . . . . . . 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 264 . . . . . . . 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 1888 . . . . . . 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 735 . . . . . 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 1939 . . . . . 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 267 . . . . 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 732 . . . 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 264 . . 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 1915 . 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 220 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 196  wo 382  wa 383  w3a 1072  wal 1622  wex 1845  wcel 2131  ∃*wmo 2600  wral 3042  wrex 3043  Vcvv 3332  cdif 3704  cin 3706  wss 3707  c0 4050  {cpr 4315   class class class wbr 4796  cdom 8111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-reg 8654  ax-inf2 8703  ax-cc 9441  ax-ac2 9469  ax-groth 9829
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-oi 8572  df-card 8947  df-acn 8950  df-ac 9121  df-cda 9174
This theorem is referenced by: (None)
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