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Theorem grothprim 10871
Description: The Tarski-Grothendieck Axiom ax-groth 10860 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 10869 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))
2 3anass 1094 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦))) ↔ (𝑥𝑦 ∧ (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))))
3 df-ss 3979 . . . . . . . . . . . . 13 (𝑤𝑧 ↔ ∀𝑢(𝑢𝑤𝑢𝑧))
4 elin 3978 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦𝑣) ↔ (𝑤𝑦𝑤𝑣))
53, 4imbi12i 350 . . . . . . . . . . . 12 ((𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ (∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
65albii 1815 . . . . . . . . . . 11 (∀𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
76rexbii 3091 . . . . . . . . . 10 (∃𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∃𝑣𝑦𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))
8 df-rex 3068 . . . . . . . . . 10 (∃𝑣𝑦𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)) ↔ ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
97, 8bitri 275 . . . . . . . . 9 (∃𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
109ralbii 3090 . . . . . . . 8 (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑧𝑦𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))))
11 df-ral 3059 . . . . . . . 8 (∀𝑧𝑦𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣))) ↔ ∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))))
1210, 11bitri 275 . . . . . . 7 (∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ↔ ∀𝑧(𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))))
13 df-ss 3979 . . . . . . . . . . 11 (𝑧𝑦 ↔ ∀𝑤(𝑤𝑧𝑤𝑦))
14 vex 3481 . . . . . . . . . . . . . . 15 𝑦 ∈ V
1514difexi 5335 . . . . . . . . . . . . . 14 (𝑦𝑧) ∈ V
16 vex 3481 . . . . . . . . . . . . . 14 𝑧 ∈ V
17 disjdifr 4478 . . . . . . . . . . . . . 14 ((𝑦𝑧) ∩ 𝑧) = ∅
1815, 16, 17brdom6disj 10569 . . . . . . . . . . . . 13 ((𝑦𝑧) ≼ 𝑧 ↔ ∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤))
1918orbi1i 913 . . . . . . . . . . . 12 (((𝑦𝑧) ≼ 𝑧𝑧𝑦) ↔ (∃𝑤(∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
20 19.44v 1989 . . . . . . . . . . . 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))
2119, 20bitr4i 278 . . . . . . . . . . 11 (((𝑦𝑧) ≼ 𝑧𝑧𝑦) ↔ ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦))
2213, 21imbi12i 350 . . . . . . . . . 10 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ (∀𝑤(𝑤𝑧𝑤𝑦) → ∃𝑤((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
23 19.35 1874 . . . . . . . . . 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)))
2422, 23bitr4i 278 . . . . . . . . 9 ((𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)) ↔ ∃𝑤((𝑤𝑧𝑤𝑦) → ((∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ∧ ∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ∨ 𝑧𝑦)))
25 grothprimlem 10870 . . . . . . . . . . . . . . . . . 18 ({𝑣, 𝑢} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))))
2625mobii 2545 . . . . . . . . . . . . . . . . 17 (∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∃*𝑢𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))))
27 df-mo 2537 . . . . . . . . . . . . . . . . 17 (∃*𝑢𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) ↔ ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
2826, 27bitri 275 . . . . . . . . . . . . . . . 16 (∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
2928ralbii 3090 . . . . . . . . . . . . . . 15 (∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∀𝑣𝑧𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡))
30 df-ral 3059 . . . . . . . . . . . . . . 15 (∀𝑣𝑧𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡) ↔ ∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)))
3129, 30bitri 275 . . . . . . . . . . . . . 14 (∀𝑣𝑧 ∃*𝑢{𝑣, 𝑢} ∈ 𝑤 ↔ ∀𝑣(𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)))
32 df-ral 3059 . . . . . . . . . . . . . . 15 (∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∀𝑣(𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤))
33 eldif 3972 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝑦𝑧) ↔ (𝑣𝑦 ∧ ¬ 𝑣𝑧))
34 grothprimlem 10870 . . . . . . . . . . . . . . . . . . . 20 ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
3534rexbii 3091 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑢𝑧𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
36 df-rex 3068 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑧𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))) ↔ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))
3735, 36bitri 275 . . . . . . . . . . . . . . . . . 18 (∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))
3833, 37imbi12i 350 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ((𝑣𝑦 ∧ ¬ 𝑣𝑧) → ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))
39 pm5.6 1003 . . . . . . . . . . . . . . . . 17 (((𝑣𝑦 ∧ ¬ 𝑣𝑧) → ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))) ↔ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4038, 39bitri 275 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4140albii 1815 . . . . . . . . . . . . . . 15 (∀𝑣(𝑣 ∈ (𝑦𝑧) → ∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤) ↔ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4232, 41bitri 275 . . . . . . . . . . . . . 14 (∀𝑣 ∈ (𝑦𝑧)∃𝑢𝑧 {𝑢, 𝑣} ∈ 𝑤 ↔ ∀𝑣(𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣)))))))
4331, 42anbi12i 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))))))))
44 19.26 1867 . . . . . . . . . . . . 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))))))))
4543, 44bitr4i 278 . . . . . . . . . . . 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))))))))
4645orbi1i 913 . . . . . . . . . . 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))
4746imbi2i 336 . . . . . . . . . 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)))
4847exbii 1844 . . . . . . . . 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)))
4924, 48bitri 275 . . . . . . . 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)))
5049albii 1815 . . . . . . 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)))
5112, 50anbi12i 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))))
52 19.26 1867 . . . . . 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))))
5351, 52bitr4i 278 . . . . 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))))
5453anbi2i 623 . . . 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)))))
552, 54bitri 275 . . 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)))))
5655exbii 1844 . 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)))))
571, 56mpbi 230 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 206  wa 395  wo 847  w3a 1086  wal 1534  wex 1775  wcel 2105  ∃*wmo 2535  wral 3058  wrex 3067  cdif 3959  cin 3961  wss 3962  {cpr 4632   class class class wbr 5147  cdom 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-reg 9629  ax-inf2 9678  ax-cc 10472  ax-ac2 10500  ax-groth 10860
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-oi 9547  df-dju 9938  df-card 9976  df-acn 9979  df-ac 10153
This theorem is referenced by: (None)
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