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Theorem finomni 6699
Description: A finite set is omniscient. Remark right after Definition 3.1 of [Pierik], p. 14. (Contributed by Jim Kingdon, 28-Jun-2022.)
Assertion
Ref Expression
finomni (𝐴 ∈ Fin → 𝐴 ∈ Omni)

Proof of Theorem finomni
Dummy variables 𝑤 𝑦 𝑧 𝑓 𝑔 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2145 . 2 (𝑤 = ∅ → (𝑤 ∈ Omni ↔ ∅ ∈ Omni))
2 eleq1 2145 . 2 (𝑤 = 𝑦 → (𝑤 ∈ Omni ↔ 𝑦 ∈ Omni))
3 eleq1 2145 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ∈ Omni ↔ (𝑦 ∪ {𝑧}) ∈ Omni))
4 eleq1 2145 . 2 (𝑤 = 𝐴 → (𝑤 ∈ Omni ↔ 𝐴 ∈ Omni))
5 0ex 3931 . . . 4 ∅ ∈ V
6 isomni 6695 . . . 4 (∅ ∈ V → (∅ ∈ Omni ↔ ∀𝑓(𝑓:∅⟶2𝑜 → (∃𝑥 ∈ ∅ (𝑓𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓𝑥) = 1𝑜))))
75, 6ax-mp 7 . . 3 (∅ ∈ Omni ↔ ∀𝑓(𝑓:∅⟶2𝑜 → (∃𝑥 ∈ ∅ (𝑓𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓𝑥) = 1𝑜)))
8 ral0 3364 . . . . 5 𝑥 ∈ ∅ (𝑓𝑥) = 1𝑜
98olci 684 . . . 4 (∃𝑥 ∈ ∅ (𝑓𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓𝑥) = 1𝑜)
109a1i 9 . . 3 (𝑓:∅⟶2𝑜 → (∃𝑥 ∈ ∅ (𝑓𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓𝑥) = 1𝑜))
117, 10mpgbir 1383 . 2 ∅ ∈ Omni
12 elun1 3151 . . . . . . . . . . . 12 (𝑥𝑦𝑥 ∈ (𝑦 ∪ {𝑧}))
1312ad2antlr 473 . . . . . . . . . . 11 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → 𝑥 ∈ (𝑦 ∪ {𝑧}))
14 fvres 5272 . . . . . . . . . . . . 13 (𝑥𝑦 → ((𝑔𝑦)‘𝑥) = (𝑔𝑥))
1514ad2antlr 473 . . . . . . . . . . . 12 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → ((𝑔𝑦)‘𝑥) = (𝑔𝑥))
16 simpr 108 . . . . . . . . . . . 12 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → ((𝑔𝑦)‘𝑥) = ∅)
1715, 16eqtr3d 2117 . . . . . . . . . . 11 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → (𝑔𝑥) = ∅)
18 fveq2 5251 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑔𝑢) = (𝑔𝑥))
1918eqeq1d 2091 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑔𝑢) = ∅ ↔ (𝑔𝑥) = ∅))
2019rspcev 2712 . . . . . . . . . . 11 ((𝑥 ∈ (𝑦 ∪ {𝑧}) ∧ (𝑔𝑥) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅)
2113, 17, 20syl2anc 403 . . . . . . . . . 10 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅)
2221orcd 685 . . . . . . . . 9 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜))
2322ex 113 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ 𝑥𝑦) → (((𝑔𝑦)‘𝑥) = ∅ → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜)))
2423rexlimdva 2483 . . . . . . 7 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜)))
25 vsnid 3450 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
26 elun2 3152 . . . . . . . . . . . . 13 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
2725, 26ax-mp 7 . . . . . . . . . . . 12 𝑧 ∈ (𝑦 ∪ {𝑧})
2827a1i 9 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
29 fveq2 5251 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑔𝑢) = (𝑔𝑧))
3029eqeq1d 2091 . . . . . . . . . . . 12 (𝑢 = 𝑧 → ((𝑔𝑢) = ∅ ↔ (𝑔𝑧) = ∅))
3130rspcev 2712 . . . . . . . . . . 11 ((𝑧 ∈ (𝑦 ∪ {𝑧}) ∧ (𝑔𝑧) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅)
3228, 31sylan 277 . . . . . . . . . 10 ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ (𝑔𝑧) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅)
3332orcd 685 . . . . . . . . 9 ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ (𝑔𝑧) = ∅) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜))
3433a1d 22 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ (𝑔𝑧) = ∅) → (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜 → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜)))
35 simpr 108 . . . . . . . . . . . 12 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ (𝑔𝑧) = 1𝑜) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜) → ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜)
36 fveq2 5251 . . . . . . . . . . . . . . 15 (𝑥 = 𝑢 → ((𝑔𝑦)‘𝑥) = ((𝑔𝑦)‘𝑢))
3736eqeq1d 2091 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (((𝑔𝑦)‘𝑥) = 1𝑜 ↔ ((𝑔𝑦)‘𝑢) = 1𝑜))
3837cbvralv 2583 . . . . . . . . . . . . 13 (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜 ↔ ∀𝑢𝑦 ((𝑔𝑦)‘𝑢) = 1𝑜)
39 fvres 5272 . . . . . . . . . . . . . . 15 (𝑢𝑦 → ((𝑔𝑦)‘𝑢) = (𝑔𝑢))
4039eqeq1d 2091 . . . . . . . . . . . . . 14 (𝑢𝑦 → (((𝑔𝑦)‘𝑢) = 1𝑜 ↔ (𝑔𝑢) = 1𝑜))
4140ralbiia 2386 . . . . . . . . . . . . 13 (∀𝑢𝑦 ((𝑔𝑦)‘𝑢) = 1𝑜 ↔ ∀𝑢𝑦 (𝑔𝑢) = 1𝑜)
4238, 41bitri 182 . . . . . . . . . . . 12 (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜 ↔ ∀𝑢𝑦 (𝑔𝑢) = 1𝑜)
4335, 42sylib 120 . . . . . . . . . . 11 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ (𝑔𝑧) = 1𝑜) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜) → ∀𝑢𝑦 (𝑔𝑢) = 1𝑜)
44 simplr 497 . . . . . . . . . . . 12 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ (𝑔𝑧) = 1𝑜) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜) → (𝑔𝑧) = 1𝑜)
45 vex 2615 . . . . . . . . . . . . 13 𝑧 ∈ V
4629eqeq1d 2091 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → ((𝑔𝑢) = 1𝑜 ↔ (𝑔𝑧) = 1𝑜))
4745, 46ralsn 3460 . . . . . . . . . . . 12 (∀𝑢 ∈ {𝑧} (𝑔𝑢) = 1𝑜 ↔ (𝑔𝑧) = 1𝑜)
4844, 47sylibr 132 . . . . . . . . . . 11 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ (𝑔𝑧) = 1𝑜) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜) → ∀𝑢 ∈ {𝑧} (𝑔𝑢) = 1𝑜)
49 ralun 3166 . . . . . . . . . . 11 ((∀𝑢𝑦 (𝑔𝑢) = 1𝑜 ∧ ∀𝑢 ∈ {𝑧} (𝑔𝑢) = 1𝑜) → ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜)
5043, 48, 49syl2anc 403 . . . . . . . . . 10 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ (𝑔𝑧) = 1𝑜) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜) → ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜)
5150olcd 686 . . . . . . . . 9 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ (𝑔𝑧) = 1𝑜) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜))
5251ex 113 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) ∧ (𝑔𝑧) = 1𝑜) → (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜 → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜)))
53 simpr 108 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) → 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜)
5453, 28ffvelrnd 5378 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) → (𝑔𝑧) ∈ 2𝑜)
55 df2o3 6125 . . . . . . . . . 10 2𝑜 = {∅, 1𝑜}
5654, 55syl6eleq 2175 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) → (𝑔𝑧) ∈ {∅, 1𝑜})
57 elpri 3445 . . . . . . . . 9 ((𝑔𝑧) ∈ {∅, 1𝑜} → ((𝑔𝑧) = ∅ ∨ (𝑔𝑧) = 1𝑜))
5856, 57syl 14 . . . . . . . 8 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) → ((𝑔𝑧) = ∅ ∨ (𝑔𝑧) = 1𝑜))
5934, 52, 58mpjaodan 745 . . . . . . 7 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) → (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜 → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜)))
60 vex 2615 . . . . . . . . . . . 12 𝑦 ∈ V
61 isomni 6695 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑦 ∈ Omni ↔ ∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜))))
6260, 61ax-mp 7 . . . . . . . . . . 11 (𝑦 ∈ Omni ↔ ∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜)))
6362biimpi 118 . . . . . . . . . 10 (𝑦 ∈ Omni → ∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜)))
6463adantl 271 . . . . . . . . 9 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜)))
65 vex 2615 . . . . . . . . . . 11 𝑔 ∈ V
6665resex 4708 . . . . . . . . . 10 (𝑔𝑦) ∈ V
67 feq1 5096 . . . . . . . . . . 11 (𝑓 = (𝑔𝑦) → (𝑓:𝑦⟶2𝑜 ↔ (𝑔𝑦):𝑦⟶2𝑜))
68 fveq1 5250 . . . . . . . . . . . . . 14 (𝑓 = (𝑔𝑦) → (𝑓𝑥) = ((𝑔𝑦)‘𝑥))
6968eqeq1d 2091 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝑦) → ((𝑓𝑥) = ∅ ↔ ((𝑔𝑦)‘𝑥) = ∅))
7069rexbidv 2375 . . . . . . . . . . . 12 (𝑓 = (𝑔𝑦) → (∃𝑥𝑦 (𝑓𝑥) = ∅ ↔ ∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅))
7168eqeq1d 2091 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝑦) → ((𝑓𝑥) = 1𝑜 ↔ ((𝑔𝑦)‘𝑥) = 1𝑜))
7271ralbidv 2374 . . . . . . . . . . . 12 (𝑓 = (𝑔𝑦) → (∀𝑥𝑦 (𝑓𝑥) = 1𝑜 ↔ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜))
7370, 72orbi12d 740 . . . . . . . . . . 11 (𝑓 = (𝑔𝑦) → ((∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜) ↔ (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ ∨ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜)))
7467, 73imbi12d 232 . . . . . . . . . 10 (𝑓 = (𝑔𝑦) → ((𝑓:𝑦⟶2𝑜 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜)) ↔ ((𝑔𝑦):𝑦⟶2𝑜 → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ ∨ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜))))
7566, 74spcv 2702 . . . . . . . . 9 (∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜)) → ((𝑔𝑦):𝑦⟶2𝑜 → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ ∨ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜)))
7664, 75syl 14 . . . . . . . 8 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ((𝑔𝑦):𝑦⟶2𝑜 → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ ∨ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜)))
77 ssun1 3147 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
78 fssres 5133 . . . . . . . . 9 ((𝑔:(𝑦 ∪ {𝑧})⟶2𝑜𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑔𝑦):𝑦⟶2𝑜)
7977, 78mpan2 416 . . . . . . . 8 (𝑔:(𝑦 ∪ {𝑧})⟶2𝑜 → (𝑔𝑦):𝑦⟶2𝑜)
8076, 79impel 274 . . . . . . 7 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ ∨ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1𝑜))
8124, 59, 80mpjaod 671 . . . . . 6 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2𝑜) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜))
8281ex 113 . . . . 5 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → (𝑔:(𝑦 ∪ {𝑧})⟶2𝑜 → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜)))
8382alrimiv 1797 . . . 4 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2𝑜 → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜)))
8445snex 3983 . . . . . 6 {𝑧} ∈ V
8560, 84unex 4229 . . . . 5 (𝑦 ∪ {𝑧}) ∈ V
86 isomni 6695 . . . . 5 ((𝑦 ∪ {𝑧}) ∈ V → ((𝑦 ∪ {𝑧}) ∈ Omni ↔ ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2𝑜 → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜))))
8785, 86ax-mp 7 . . . 4 ((𝑦 ∪ {𝑧}) ∈ Omni ↔ ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2𝑜 → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1𝑜)))
8883, 87sylibr 132 . . 3 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → (𝑦 ∪ {𝑧}) ∈ Omni)
8988ex 113 . 2 (𝑦 ∈ Fin → (𝑦 ∈ Omni → (𝑦 ∪ {𝑧}) ∈ Omni))
901, 2, 3, 4, 11, 89findcard2 6533 1 (𝐴 ∈ Fin → 𝐴 ∈ Omni)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  wal 1283   = wceq 1285  wcel 1434  wral 2353  wrex 2354  Vcvv 2612  cun 2982  wss 2984  c0 3269  {csn 3422  {cpr 3423  cres 4401  wf 4963  cfv 4967  1𝑜c1o 6104  2𝑜c2o 6105  Fincfn 6385  Omnicomni 6693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-1o 6111  df-2o 6112  df-er 6220  df-en 6386  df-fin 6388  df-omni 6694
This theorem is referenced by: (None)
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